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257 1 {CSTYLE "" -1 -1 "Times" 1 12 0 0 0 1 2 1 2 2 2 2 1 1 1 1 }1 1 0 0 0 0 1 0 1 0 2 2 0 1 }{PSTYLE "Normal " -1 258 1 {CSTYLE "" -1 -1 "Times" 1 12 0 0 0 1 2 2 2 2 2 2 1 1 1 1 } 1 1 0 0 0 0 1 0 1 0 2 2 0 1 }{PSTYLE "Normal" -1 259 1 {CSTYLE "" -1 -1 "Times" 1 12 0 0 0 1 2 2 2 2 2 2 1 1 1 1 }3 1 0 0 0 0 1 0 1 0 2 2 0 1 }} {SECT 0 {PARA 3 "" 0 "" {TEXT -1 20 "Integration by parts" }}{PARA 0 " " 0 "" {TEXT -1 37 "by Peter Stone, Nanaimo, B.C., Canada" }}{PARA 0 " " 0 "" {TEXT -1 19 "Version: 23.3.2007 " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 8 "restart;" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 33 "The integration by parts formula " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{PARA 0 "" 0 "" {TEXT -1 3 "If " }{XPPEDIT 18 0 "f(x) = u(x)*v(x);" "6#/-%\"fG6#%\"xG*&-%\"uG6#F'\"\"\"-%\"vG6#F'F," } {TEXT -1 8 ", where " }{XPPEDIT 18 0 "u(x)" "6#-%\"uG6#%\"xG" }{TEXT -1 5 " and " }{XPPEDIT 18 0 "v(x)" "6#-%\"vG6#%\"xG" }{TEXT -1 76 " ar e two differentiable functions, the product rule for differentiation i s: " }}{PARA 259 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "`f '`(x)=`u '` (x)*v(x)+u(x)*`v '`(x)" "6#/-%$f~'G6#%\"xG,&*&-%$u~'G6#F'\"\"\"-%\"vG6 #F'F-F-*&-%\"uG6#F'F--%$v~'G6#F'F-F-" }{TEXT -1 2 ", " }}{PARA 0 "" 0 "" {TEXT -1 2 "or" }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 " d/dx;" "6#*&%\"dG\"\"\"%#dxG!\"\"" }{TEXT -1 3 " [ " }{XPPEDIT 18 0 "u *v" "6#*&%\"uG\"\"\"%\"vGF%" }{TEXT -1 5 " ] = " }{TEXT 263 1 "u" } {TEXT -1 1 " " }{XPPEDIT 18 0 "dv/dx;" "6#*&%#dvG\"\"\"%#dxG!\"\"" } {TEXT -1 3 " + " }{TEXT 264 1 "v" }{TEXT -1 1 " " }{XPPEDIT 18 0 "du/d x;" "6#*&%#duG\"\"\"%#dxG!\"\"" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 13 "For example, " }{XPPEDIT 18 0 "d/dx;" "6#*&%\"dG\"\"\"%#d xG!\"\"" }{TEXT -1 2 "[ " }{XPPEDIT 18 0 "x^2*sin*x;" "6#*(%\"xG\"\"#% $sinG\"\"\"F$F'" }{TEXT -1 5 " ] = " }{XPPEDIT 18 0 "2*x*sin*x;" "6#** \"\"#\"\"\"%\"xGF%%$sinGF%F&F%" }{TEXT -1 3 " + " }{XPPEDIT 18 0 "x^2* cos*x;" "6#*(%\"xG\"\"#%$cosG\"\"\"F$F'" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 19 "diff(x^2*sin(x),x);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,&*&%\"xG\"\"\"-%$sinG6#F%F&\"\"#*&)F%F*F&-%$cosGF)F&F& " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 24 "Integrating the formula \+ " }{XPPEDIT 18 0 "d/dx;" "6#*&%\"dG\"\"\"%#dxG!\"\"" }{TEXT -1 3 " [ \+ " }{XPPEDIT 18 0 "u*v" "6#*&%\"uG\"\"\"%\"vGF%" }{TEXT -1 5 " ] = " } {TEXT 265 1 "u" }{TEXT -1 1 " " }{XPPEDIT 18 0 "dv/dx;" "6#*&%#dvG\"\" \"%#dxG!\"\"" }{TEXT -1 3 " + " }{TEXT 266 1 "v" }{TEXT -1 1 " " } {XPPEDIT 18 0 "du/dx;" "6#*&%#duG\"\"\"%#dxG!\"\"" }{TEXT -1 18 ", wit h respect to " }{TEXT 267 1 "x" }{TEXT -1 7 " gives:" }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "u*v =Int(u*``(dv/dx),x) +Int(v*``( du/dx),x)" "6#/*&%\"uG\"\"\"%\"vGF&,&-%$IntG6$*&F%F&-%!G6#*&%#dvGF&%#d xG!\"\"F&%\"xGF&-F*6$*&F'F&-F.6#*&%#duGF&F2F3F&F4F&" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 37 "Re-arranging this equation we obtain:" }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "Int(u*``(dv/dx),x)= u*v - Int(v*``(du/dx),x)" "6#/-%$IntG6$*&%\"uG\"\"\"-%!G6#*&%#dvGF)%#d xG!\"\"F)%\"xG,&*&F(F)%\"vGF)F)-F%6$*&F4F)-F+6#*&%#duGF)F/F0F)F1F0" } {TEXT -1 2 ", " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{TEXT 262 19 "____ _______________" }{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 13 "which \+ is the " }{TEXT 259 28 "integration by parts formula" }{TEXT -1 2 ". \+ " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 4 "The " }{TEXT 0 7 "student" }{TEXT -1 30 " package contains a procedure " } {TEXT 0 8 "intparts" }{TEXT -1 88 ", which applies the integration by \+ parts formula to a specified integral. The procedure " }{TEXT 0 8 "int parts" }{TEXT -1 123 " takes the integral it is to be applied to as it s first argument, and the second argument is the expression to be take n as " }{TEXT 277 1 "u" }{TEXT -1 17 " in the formula. " }}{PARA 0 "" 0 "" {TEXT -1 90 "The procedure intparts takes two arguments. The firs t argument is an integral of the form " }{TEXT 268 18 "Int(u(x)*dv/dx, x)" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 34 "The second argume nt is the factor " }{XPPEDIT 18 0 "u=u(x)" "6#/%\"uG-F$6#%\"xG" } {TEXT -1 16 " in the formula " }{XPPEDIT 18 0 "Int(u*``(dv/dx),x) = u* v-Int(v*``(du/dx),x)" "6#/-%$IntG6$*&%\"uG\"\"\"-%!G6#*&%#dvGF)%#dxG! \"\"F)%\"xG,&*&F(F)%\"vGF)F)-F%6$*&F4F)-F+6#*&%#duGF)F/F0F)F1F0" } {TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 72 "u := 'u': v := 'v':\nInt(u(x)*Diff(v(x),x),x);\n student[intparts](%,u(x));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%$IntG6 $*&-%\"uG6#%\"xG\"\"\"-%%DiffG6$-%\"vGF)F*F+F*" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,&*&-%\"uG6#%\"xG\"\"\"-%\"vGF'F)F)-%$IntG6$*&-%%diffG6 $F%F(F)F*F)F(!\"\"" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 49 " A procedure for performing integration by parts: " }{TEXT 0 10 "intbyp arts" }{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 17 "intbyparts: usage" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }{TEXT 300 18 "Calling Sequence:\n" }}{PARA 0 "" 0 "" {TEXT -1 29 " intbyparts( int, u=ux, v ) " }}{PARA 0 "" 0 "" {TEXT -1 32 " intbyparts( int, u=ux, v=vx ) " }{TEXT 303 1 "\n" }{TEXT -1 0 "" }}{PARA 257 "" 0 "" {TEXT -1 11 "Parameters:" }}{PARA 0 "" 0 "" {TEXT -1 5 " " }}{PARA 0 "" 0 "" {TEXT 23 10 " int - " }{TEXT -1 42 " an indefinite integral Int(fx,x), say x." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 4 " " }{TEXT 23 9 "u=ux - " }{TEXT -1 105 "the factor of the integrand fx in the first argumen t to take for u in the integration by parts formula: " }}{PARA 0 "" 0 "" {TEXT -1 69 " Int(u*diff(v,x),x)=u*v-In t(v*diff(u,x),x)." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 7 " " }{TEXT 23 8 "v - " }{TEXT -1 45 "name for the 2nd factor in the first term u*v" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {PARA 0 "" 0 "" {TEXT -1 4 " " }{TEXT 23 9 "v=vx - " }{TEXT -1 15 "the 2nd factor " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 257 "" 0 "" {TEXT -1 12 "Description:" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 14 "The procedure " }{TEXT 0 10 "intbyparts" } {TEXT -1 51 " attempts to apply the integration by parts formula" }} {PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "Int(u*``(dv/dx),x)=u* v - Int(v*``(du/dx),x)" "6#/-%$IntG6$*&%\"uG\"\"\"-%!G6#*&%#dvGF)%#dxG !\"\"F)%\"xG,&*&F(F)%\"vGF)F)-F%6$*&F4F)-F+6#*&%#duGF)F/F0F)F1F0" } {TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 25 "to an idefinite integral " }}{PARA 0 "" 0 "" {TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT 301 8 "Opt ions:" }{TEXT -1 1 "\n" }}{PARA 0 "" 0 "" {TEXT -1 85 "info=true/false \nWith the option \"info=true\" some of the steps involved will be sho wn." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 259 16 "How to activate:" }{TEXT 256 1 "\n" } {TEXT -1 155 "To make the procedure active, open the subsection, place the cursor anywhere after the prompt [ > and press [Enter].\nYou can then close up the subsection." }}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 27 "intbyparts: implementation " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 2108 "intbyparts := proc(ff,uu) \n local fx,x,ux,vx,vv,dux,dvx,u,v,d,prntflg,opt;\n \n prntfl g := false;\n if nargs>2 then\n vv := args[3];\n if type(v v,`=`) and type(op(1,vv),name) and op(1,vv)<>'info' then\n v : = op(1,vv);\n vx := op(2,vv);\n elif type(vv,'name') then \n v := vv;\n else\n error \"the 3rd argument, %1 , is invalid .. it should give (the name for) the '2nd' factor of the \+ product to be constructed\",vv;\n end if;\n end if;\n if n args>3 then\n opt := args[4];\n if type(opt,`=`) and op(1,op t)='info' then\n prntflg := op(2,opt):\n if prntflg<>t rue then prntflg := false end if;\n else\n error \"the 4t h argument, %1, is invalid .. it should be \\\"info=true\\\" or \\\"in fo=false\\\"\",opt;\n end if;\n end if; \n\n if type(ff,' function') and member(op(0,ff),\{'Int','int'\}) then\n fx := op(1 ,ff);\n x := op(2,ff);\n if type(uu,'name'='algebraic') then \n u := op(1,uu);\n ux := op(2,uu);\n elif type(u u,algebraic) and not prntflg then\n ux := uu;\n else\n \+ if prntflg then\n error \"the 2nd argument, %1, is in valid .. it should have the form 'u=expr' to give the factor to be dif ferentiated with respect to %2\",uu,x;\n else\n err or \"the 2nd argument, %1, is invalid .. it should give the factor to \+ be differentiated with respect to %2\",uu,x;\n end if;\n \+ end if;\n\n dvx := traperror(normal(fx/ux));\n if dvx=laster ror then error \"%1 is not a factor of %2\",ux,fx end if;\n dux : = diff(ux,x);\n if assigned(vx) then\n if diff(vx,x)<>dvx then\n error \"the 2nd argument, %1, is invalid .. the der ivative of %2 with respect to %3 should be %4\",vv,vx,x,dvx;\n \+ end if;\n else\n vx := int(dvx,x);\n end if;\n \+ if prntflg then\n print(PIECEWISE([u=ux,v=vx],[(`d`*u)/(` d`* x)=dux,(`d`*v)/(` d`*x)=dvx]));\n print(ff=Int(u*``((`d`*v)/(` d`*x)),x));\n print(``=u*v-Int(v*``((`d`*u)/(` d`*x)),x));\n \+ end if;\n ux*vx-Int(vx*dux,x);\n end if;\nend proc:" }}} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 39 "Examples are given in the next section." }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 52 "Examples i nvolving the integration by parts formula " }}{EXCHG {PARA 0 "> " 0 " " {MPLTEXT 1 0 1 ";" }}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 9 "Example 1 " }}{PARA 258 "" 0 "" {TEXT 273 8 "Question" }{TEXT -1 10 ": Find \+ " }{XPPEDIT 18 0 "Int(x*cos*x,x);" "6#-%$IntG6$*(%\"xG\"\"\"%$cosGF(F' F(F'" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT 274 8 "Solution" } {TEXT -1 2 ": " }}{PARA 0 "" 0 "" {TEXT -1 4 "Let " }{XPPEDIT 18 0 "u \+ = x" "6#/%\"uG%\"xG" }{TEXT -1 5 " and " }{XPPEDIT 18 0 "v = sin*x;" " 6#/%\"vG*&%$sinG\"\"\"%\"xGF'" }{TEXT -1 38 " in the integration by pa rts formula. " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "Int( x*cos*x,x);" "6#-%$IntG6$*(%\"xG\"\"\"%$cosGF(F'F(F'" }{TEXT -1 7 " .. . " }{XPPEDIT 18 0 "PIECEWISE([u = x, v = sin*x],[du/dx = 1, dv/dx = cos*x]);" "6#-%*PIECEWISEG6$7$/%\"uG%\"xG/%\"vG*&%$sinG\"\"\"F)F.7$/* &%#duGF.%#dxG!\"\"F./*&%#dvGF.F3F4*&%$cosGF.F)F." }{TEXT -1 1 " " }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 256 "" 0 "" {TEXT -1 1 " " } {XPPEDIT 18 0 "``= Int(u*``(dv/dx),x)" "6#/%!G-%$IntG6$*&%\"uG\"\"\"-F $6#*&%#dvGF*%#dxG!\"\"F*%\"xG" }{TEXT -1 1 " " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "``= u*v-Int(v*``(du/dx),x)" "6#/%!G,&*& %\"uG\"\"\"%\"vGF(F(-%$IntG6$*&F)F(-F$6#*&%#duGF(%#dxG!\"\"F(%\"xGF3" }{TEXT -1 1 " " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "`` \+ = x*sin*x-Int(sin*x,x);" "6#/%!G,&*(%\"xG\"\"\"%$sinGF(F'F(F(-%$IntG6$ *&F)F(F'F(F'!\"\"" }{TEXT -1 1 " " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "`` = x*sin*x+cos*x+c;" "6#/%!G,(*(%\"xG\"\"\"%$sinGF(F 'F(F(*&%$cosGF(F'F(F(%\"cGF(" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 31 "Int(x*cos(x) ,x);\n``=value(%)+c;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%$IntG6$*&%\" xG\"\"\"-%$cosG6#F'F(F'" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/%!G,(-%$co sG6#%\"xG\"\"\"*&-%$sinGF(F*F)F*F*%\"cGF*" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 25 "We can use the procedure " } {TEXT 0 8 "intparts" }{TEXT -1 58 " to provide the intermediate step i n the previous example." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 63 "Int(x*cos(x),x);\n``=student[intpar ts](%,x);\n``=value(rhs(%))+c;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%$I ntG6$*&%\"xG\"\"\"-%$cosG6#F'F(F'" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/ %!G,&*&%\"xG\"\"\"-%$sinG6#F'F(F(-%$IntG6$F)F'!\"\"" }}{PARA 11 "" 1 " " {XPPMATH 20 "6#/%!G,(*&%\"xG\"\"\"-%$sinG6#F'F(F(-%$cosGF+F(%\"cGF( " }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 54 "More details are shown by using the special procedure " }{TEXT 0 10 "intby parts" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 70 "Int(x*cos(x),x);\n``=intbyparts(%,u =x,v,info=true);\n``=value(rhs(%))+c;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%$IntG6$*&%\"xG\"\"\"-%$cosG6#F'F(F'" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%*PIECEWISEG6$7$/%\"uG%\"xG/%\"vG-%$sinG6#F)7$/**%\"dG\"\"\"F(F 3%#~dG!\"\"F)F5F3/**F2F3F+F3F4F5F)F5-%$cosGF." }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%$IntG6$*&%\"xG\"\"\"-%$cosG6#F(F)F(-F%6$*&%\"uGF)-%! G6#**%\"dGF)%\"vGF)%#~dG!\"\"F(F8F)F(" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/%!G,&*&%\"uG\"\"\"%\"vGF(F(-%$IntG6$*&F)F(-F$6#**%\"dGF(F'F(%#~dG! \"\"%\"xGF3F(F4F3" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/%!G,&*&%\"xG\"\" \"-%$sinG6#F'F(F(-%$IntG6$F)F'!\"\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6 #/%!G,(*&%\"xG\"\"\"-%$sinG6#F'F(F(-%$cosGF+F(%\"cGF(" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}} {SECT 1 {PARA 4 "" 0 "" {TEXT -1 10 "Example 2 " }}{PARA 258 "" 0 "" {TEXT 271 8 "Question" }{TEXT -1 10 ": Find " }{XPPEDIT 18 0 "Int(x ^2*sin*x,x);" "6#-%$IntG6$*(%\"xG\"\"#%$sinG\"\"\"F'F*F'" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT 272 8 "Solution" }{TEXT -1 2 ": " }} {PARA 0 "" 0 "" {TEXT -1 47 "Applying the integration by parts formula with " }{XPPEDIT 18 0 "u = x^2;" "6#/%\"uG*$%\"xG\"\"#" }{TEXT -1 5 " and " }{XPPEDIT 18 0 "v = -cos*x;" "6#/%\"vG,$*&%$cosG\"\"\"%\"xGF(! \"\"" }{TEXT -1 8 " gives: " }}{PARA 256 "" 0 "" {TEXT -1 1 " " } {XPPEDIT 18 0 "Int(x^2*sin*x,x);" "6#-%$IntG6$*(%\"xG\"\"#%$sinG\"\"\" F'F*F'" }{TEXT -1 7 " ... " }{XPPEDIT 18 0 "PIECEWISE([u = x^2, v = \+ -cos*x],[du/dx = 2*x, dv/dx = sin*x]);" "6#-%*PIECEWISEG6$7$/%\"uG*$% \"xG\"\"#/%\"vG,$*&%$cosG\"\"\"F*F1!\"\"7$/*&%#duGF1%#dxGF2*&F+F1F*F1/ *&%#dvGF1F7F2*&%$sinGF1F*F1" }{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "``= Int(u*` `(dv/dx),x)" "6#/%!G-%$IntG6$*&%\"uG\"\"\"-F$6#*&%#dvGF*%#dxG!\"\"F*% \"xG" }{TEXT -1 1 " " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "`` = u*v-Int(v*``(du/dx),x)" "6#/%!G,&*&%\"uG\"\"\"%\"vGF(F(-%$IntG 6$*&F)F(-F$6#*&%#duGF(%#dxG!\"\"F(%\"xGF3" }{TEXT -1 1 " " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "`` = -x^2*cos*x-Int(``(-cos *x)*``(2*x),x);" "6#/%!G,&*(%\"xG\"\"#%$cosG\"\"\"F'F*!\"\"-%$IntG6$*& -F$6#,$*&F)F*F'F*F+F*-F$6#*&F(F*F'F*F*F'F+" }{TEXT -1 1 " " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "`` = -x^2*cos*x+2*Int(x*cos *x,x);" "6#/%!G,&*(%\"xG\"\"#%$cosG\"\"\"F'F*!\"\"*&F(F*-%$IntG6$*(F'F *F)F*F'F*F'F*F*" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 38 "The f irst example provides the result " }{XPPEDIT 18 0 "Int(x*cos*x,x) = x* sin*x+cos*x+c;" "6#/-%$IntG6$*(%\"xG\"\"\"%$cosGF)F(F)F(,(*(F(F)%$sinG F)F(F)F)*&F*F)F(F)F)%\"cGF)" }{TEXT -1 37 ", so, making use of this, w e obtain: " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "Int(x^2 *sin*x,x) = -x^2*cos*x+2*x*sin*x+2*cos*x+c[1];" "6#/-%$IntG6$*(%\"xG\" \"#%$sinG\"\"\"F(F+F(,**(F(F)%$cosGF+F(F+!\"\"**F)F+F(F+F*F+F(F+F+*(F) F+F.F+F(F+F+&%\"cG6#F+F+" }{TEXT -1 2 ", " }}{PARA 0 "" 0 "" {TEXT -1 6 "where " }{XPPEDIT 18 0 "c[1] = 2*c;" "6#/&%\"cG6#\"\"\"*&\"\"#F'F%F '" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 259 4 "Note" }{TEXT -1 16 ": The procedure " }{TEXT 0 8 "intpart s" }{TEXT -1 93 " is applied to all integrals in an expression, but do es not affect other factors or summands." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 120 "Int(x^2*sin(x),x);\n ``=student[intparts](%,x^2);\n``=simplify(rhs(%));\n``=student[intpart s](rhs(%),x);\n``=value(rhs(%))+c;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6# -%$IntG6$*&)%\"xG\"\"#\"\"\"-%$sinG6#F(F*F(" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/%!G,&*&)%\"xG\"\"#\"\"\"-%$cosG6#F(F*!\"\"-%$IntG6$,$* (F)F*F(F*F+F*F.F(F." }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/%!G,&*&)%\"xG \"\"#\"\"\"-%$cosG6#F(F*!\"\"*&F)F*-%$IntG6$*&F(F*F+F*F(F*F*" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/%!G,(*&)%\"xG\"\"#\"\"\"-%$cosG6#F(F*!\"\"* (F)F*F(F*-%$sinGF-F*F**&F)F*-%$IntG6$F0F(F*F." }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/%!G,**&)%\"xG\"\"#\"\"\"-%$cosG6#F(F*!\"\"*(F)F*F(F*-% $sinGF-F*F**&F)F*F+F*F*%\"cGF*" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {PARA 0 "" 0 "" {TEXT -1 107 "More details of the first application of the integration by parts formula are shown by using the procedure " } {TEXT 0 10 "intbyparts" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 95 "Int(x^2*sin(x),x);\n``= intbyparts(%,u=x^2,v,info=true);\n``=simplify(rhs(%));\n``=value(rhs(% ))+c;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%$IntG6$*&)%\"xG\"\"#\"\"\"- %$sinG6#F(F*F(" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%*PIECEWISEG6$7$/% \"uG*$)%\"xG\"\"#\"\"\"/%\"vG,$-%$cosG6#F+!\"\"7$/**%\"dGF-F(F-%#~dGF4 F+F4,$*&F,F-F+F-F-/**F8F-F/F-F9F4F+F4-%$sinGF3" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%$IntG6$*&)%\"xG\"\"#\"\"\"-%$sinG6#F)F+F)-F%6$*&%\"u GF+-%!G6#**%\"dGF+%\"vGF+%#~dG!\"\"F)F:F+F)" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/%!G,&*&%\"uG\"\"\"%\"vGF(F(-%$IntG6$*&F)F(-F$6#**%\"dG F(F'F(%#~dG!\"\"%\"xGF3F(F4F3" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/%!G, &*&)%\"xG\"\"#\"\"\"-%$cosG6#F(F*!\"\"-%$IntG6$,$*(F)F*F(F*F+F*F.F(F. " }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/%!G,&*&)%\"xG\"\"#\"\"\"-%$cosG6# F(F*!\"\"*&F)F*-%$IntG6$*&F(F*F+F*F(F*F*" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/%!G,**&)%\"xG\"\"#\"\"\"-%$cosG6#F(F*!\"\"*&F)F*F+F*F**(F)F*F(F *-%$sinGF-F*F*%\"cGF*" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 10 "Example 3 " }}{PARA 258 "" 0 "" {TEXT 269 8 "Question" } {TEXT -1 9 ": Find " }{XPPEDIT 18 0 "Int(x*exp(2*x),x)" "6#-%$IntG6$ *&%\"xG\"\"\"-%$expG6#*&\"\"#F(F'F(F(F'" }{TEXT -1 2 ". " }}{PARA 0 " " 0 "" {TEXT 270 8 "Solution" }{TEXT -1 3 ": " }}{PARA 0 "" 0 "" {TEXT -1 47 "Applying the integration by parts formula with " } {XPPEDIT 18 0 "u = x;" "6#/%\"uG%\"xG" }{TEXT -1 5 " and " }{XPPEDIT 18 0 "v = 1/2;" "6#/%\"vG*&\"\"\"F&\"\"#!\"\"" }{TEXT -1 1 " " } {XPPEDIT 18 0 "exp(2*x)" "6#-%$expG6#*&\"\"#\"\"\"%\"xGF(" }{TEXT -1 8 " gives: " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "Int(x* exp(2*x),x);" "6#-%$IntG6$*&%\"xG\"\"\"-%$expG6#*&\"\"#F(F'F(F(F'" } {TEXT -1 6 " ... " }{XPPEDIT 18 0 "PIECEWISE([u=x,v=exp(2*x)/2],[du/d x=1,dv/dx=exp(2*x)])" "6#-%*PIECEWISEG6$7$/%\"uG%\"xG/%\"vG*&-%$expG6# *&\"\"#\"\"\"F)F2F2F1!\"\"7$/*&%#duGF2%#dxGF3F2/*&%#dvGF2F8F3-F.6#*&F1 F2F)F2" }{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 256 " " 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "``= Int(u*``(dv/dx),x)" "6#/%!G- %$IntG6$*&%\"uG\"\"\"-F$6#*&%#dvGF*%#dxG!\"\"F*%\"xG" }{TEXT -1 1 " " }}{PARA 256 "" 0 "" {TEXT -1 2 " " }{XPPEDIT 18 0 "`` = u*v-Int(v*``( du/dx),x)" "6#/%!G,&*&%\"uG\"\"\"%\"vGF(F(-%$IntG6$*&F)F(-F$6#*&%#duGF (%#dxG!\"\"F(%\"xGF3" }{TEXT -1 1 " " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "``= x/2 exp(2*x) - Int(exp(2*x)/2,x)" "6#/%!G,&*(% \"xG\"\"\"\"\"#!\"\"-%$expG6#*&F)F(F'F(F(F(-%$IntG6$*&-F,6#*&F)F(F'F(F (F)F*F'F*" }{TEXT -1 1 " " }}{PARA 256 "" 0 "" {TEXT -1 1 " " } {XPPEDIT 18 0 "``=x*exp(2*x)/2-exp(2*x)/4+c" "6#/%!G,(*(%\"xG\"\"\"-%$ expG6#*&\"\"#F(F'F(F(F-!\"\"F(*&-F*6#*&F-F(F'F(F(\"\"%F.F.%\"cGF(" } {TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 65 "Int(x*exp(2*x),x);\n``=student[intparts](%,x);\n ``=value(rhs(%))+c;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%$IntG6$*&%\"x G\"\"\"-%$expG6#,$*&\"\"#F(F'F(F(F(F'" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/%!G,&*&#\"\"\"\"\"#F(*&%\"xGF(-%$expG6#,$*&F)F(F+F(F(F(F(F(-%$IntG 6$,$*&F'F(F,F(F(F+!\"\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/%!G,(*&#\" \"\"\"\"#F(*&%\"xGF(-%$expG6#,$*&F)F(F+F(F(F(F(F(*&#F(\"\"%F(F,F(!\"\" %\"cGF(" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 46 "More details are shown by using the procedure " }{TEXT 0 10 "intby parts" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 72 "Int(x*exp(2*x),x);\n``=intbyparts(% ,u=x,v,info=true);\n``=value(rhs(%))+c;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%$IntG6$*&%\"xG\"\"\"-%$expG6#,$*&\"\"#F(F'F(F(F(F'" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%*PIECEWISEG6$7$/%\"uG%\"xG/%\"vG,$*&#\"\" \"\"\"#F/-%$expG6#,$*&F0F/F)F/F/F/F/7$/**%\"dGF/F(F/%#~dG!\"\"F)F;F//* *F9F/F+F/F:F;F)F;F1" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%$IntG6$*&%\" xG\"\"\"-%$expG6#,$*&\"\"#F)F(F)F)F)F(-F%6$*&%\"uGF)-%!G6#**%\"dGF)%\" vGF)%#~dG!\"\"F(F;F)F(" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/%!G,&*&%\"u G\"\"\"%\"vGF(F(-%$IntG6$*&F)F(-F$6#**%\"dGF(F'F(%#~dG!\"\"%\"xGF3F(F4 F3" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/%!G,&*&#\"\"\"\"\"#F(*&%\"xGF(- %$expG6#,$*&F)F(F+F(F(F(F(F(-%$IntG6$,$*&F'F(F,F(F(F+!\"\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/%!G,(*&#\"\"\"\"\"#F(*&%\"xGF(-%$expG6#,$*& F)F(F+F(F(F(F(F(*&#F(\"\"%F(F,F(!\"\"%\"cGF(" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 10 "Example 4 " }}{PARA 258 "" 0 "" {TEXT 282 8 "Question" }{TEXT -1 13 ": Find (a) " }{XPPEDIT 18 0 "Int(ln*x ,x);" "6#-%$IntG6$*&%#lnG\"\"\"%\"xGF(F)" }{TEXT -1 12 " and (b) " }{XPPEDIT 18 0 "Int((ln*x)^2,x)" "6#-%$IntG6$*$*&%#lnG\"\"\"%\"xGF)\" \"#F*" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT 283 8 "Solution" } {TEXT -1 3 ": " }}{PARA 0 "" 0 "" {TEXT -1 70 "The integration by par ts formula can be used to find these integrals. " }}{PARA 0 "" 0 "" {TEXT -1 3 "(a)" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "Int(ln*x,x);" "6#-%$IntG6$*&%#lnG\"\"\" %\"xGF(F)" }{TEXT -1 10 " ... " }{XPPEDIT 18 0 "PIECEWISE([u = ln *x, v = x],[du/dx = 1/x, dv/dx = 1]);" "6#-%*PIECEWISEG6$7$/%\"uG*&%#l nG\"\"\"%\"xGF+/%\"vGF,7$/*&%#duGF+%#dxG!\"\"*&F+F+F,F4/*&%#dvGF+F3F4F +" }{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 256 "" 0 " " {TEXT -1 1 " " }{XPPEDIT 18 0 "``= Int(u*``(dv/dx),x)" "6#/%!G-%$Int G6$*&%\"uG\"\"\"-F$6#*&%#dvGF*%#dxG!\"\"F*%\"xG" }{TEXT -1 1 " " }} {PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "`` = u*v-Int(v*``(du/ dx),x)" "6#/%!G,&*&%\"uG\"\"\"%\"vGF(F(-%$IntG6$*&F)F(-F$6#*&%#duGF(%# dxG!\"\"F(%\"xGF3" }{TEXT -1 1 " " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "`` = x*ln*x-Int(x*`.`*``(1/x),x);" "6#/%!G,&*(%\"xG\" \"\"%#lnGF(F'F(F(-%$IntG6$*(F'F(%\".GF(-F$6#*&F(F(F'!\"\"F(F'F2" } {TEXT -1 1 " " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "`` = x*ln*x-Int(1,x);" "6#/%!G,&*(%\"xG\"\"\"%#lnGF(F'F(F(-%$IntG6$F(F'!\" \"" }{TEXT -1 1 " " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "`` = x*ln*x-x+c;" "6#/%!G,(*(%\"xG\"\"\"%#lnGF(F'F(F(F'!\"\"%\"cGF(" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> \+ " 0 "" {MPLTEXT 1 0 64 "Int(ln(x),x);\n``=student[intparts](%,ln(x)); \n``=value(rhs(%))+c;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%$IntG6$-%#l nG6#%\"xGF)" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/%!G,&*&-%#lnG6#%\"xG\" \"\"F*F+F+-%$IntG6$F+F*!\"\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/%!G,( *&-%#lnG6#%\"xG\"\"\"F*F+F+F*!\"\"%\"cGF+" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 46 "More details are shown by using the procedure " }{TEXT 0 10 "intbyparts" }{TEXT -1 2 ". " }}{PARA 0 " " 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 71 "Int(l n(x),x);\n``=intbyparts(%,u=ln(x),v,info=true);\n``=value(rhs(%))+c;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%$IntG6$-%#lnG6#%\"xGF)" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%*PIECEWISEG6$7$/%\"uG-%#lnG6#%\"xG/%\"vGF, 7$/**%\"dG\"\"\"F(F3%#~dG!\"\"F,F5*&F3F3F,F5/**F2F3F.F3F4F5F,F5F3" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#/-%$IntG6$-%#lnG6#%\"xGF*-F%6$*&%\"uG \"\"\"-%!G6#**%\"dGF/%\"vGF/%#~dG!\"\"F*F7F/F*" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/%!G,&*&%\"uG\"\"\"%\"vGF(F(-%$IntG6$*&F)F(-F$6#**%\"dG F(F'F(%#~dG!\"\"%\"xGF3F(F4F3" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/%!G, &*&-%#lnG6#%\"xG\"\"\"F*F+F+-%$IntG6$F+F*!\"\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/%!G,(*&-%#lnG6#%\"xG\"\"\"F*F+F+F*!\"\"%\"cGF+" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 3 "(b)" }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "Int((ln*x)^2,x);" "6#-%$IntG6$*$*&%#lnG\"\"\" %\"xGF)\"\"#F*" }{TEXT -1 10 " ... " }{XPPEDIT 18 0 "PIECEWISE([u = (ln*x)^2, v = x],[du/dx = 2*ln*x/x, dv/dx = 1]);" "6#-%*PIECEWISEG6 $7$/%\"uG*$*&%#lnG\"\"\"%\"xGF,\"\"#/%\"vGF-7$/*&%#duGF,%#dxG!\"\"**F. F,F+F,F-F,F-F6/*&%#dvGF,F5F6F," }{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "``= I nt(u*``(dv/dx),x)" "6#/%!G-%$IntG6$*&%\"uG\"\"\"-F$6#*&%#dvGF*%#dxG!\" \"F*%\"xG" }{TEXT -1 1 " " }}{PARA 256 "" 0 "" {TEXT -1 1 " " } {XPPEDIT 18 0 "`` = u*v-Int(v*``(du/dx),x)" "6#/%!G,&*&%\"uG\"\"\"%\"v GF(F(-%$IntG6$*&F)F(-F$6#*&%#duGF(%#dxG!\"\"F(%\"xGF3" }{TEXT -1 1 " \+ " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "`` = x*(ln*x)^2-I nt(x*`.`*``(2*ln*x/x),x);" "6#/%!G,&*&%\"xG\"\"\"*$*&%#lnGF(F'F(\"\"#F (F(-%$IntG6$*(F'F(%\".GF(-F$6#**F,F(F+F(F'F(F'!\"\"F(F'F5" }{TEXT -1 1 " " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "`` = x*(ln*x) ^2-2*Int(ln*x,x);" "6#/%!G,&*&%\"xG\"\"\"*$*&%#lnGF(F'F(\"\"#F(F(*&F,F (-%$IntG6$*&F+F(F'F(F'F(!\"\"" }{TEXT -1 2 ". " }}{PARA 258 "" 0 "" {TEXT -1 6 "Since " }{XPPEDIT 18 0 "Int(ln*x,x)=x*ln*x-x+c" "6#/-%$Int G6$*&%#lnG\"\"\"%\"xGF)F*,(*(F*F)F(F)F*F)F)F*!\"\"%\"cGF)" }{TEXT -1 17 " from part (a), " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "Int((ln*x)^2,x)= x*(ln*x)^2-2*(x*ln*x-x+c)" "6#/-%$IntG6$*$*&%#l nG\"\"\"%\"xGF*\"\"#F+,&*&F+F**$*&F)F*F+F*F,F*F**&F,F*,(*(F+F*F)F*F+F* F*F+!\"\"%\"cGF*F*F4" }{TEXT -1 1 " " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "`` = x*(ln*x)^2-2*x*ln*x+2*x+c[1];" "6#/%!G,**&%\"x G\"\"\"*$*&%#lnGF(F'F(\"\"#F(F(**F,F(F'F(F+F(F'F(!\"\"*&F,F(F'F(F(&%\" cG6#F(F(" }{TEXT -1 2 ", " }}{PARA 0 "" 0 "" {TEXT -1 6 "where " } {XPPEDIT 18 0 "c[1]=2*c" "6#/&%\"cG6#\"\"\"*&\"\"#F'F%F'" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 68 "Int(ln(x)^2,x);\n``=student[intparts](%,ln(x)^2);\n`` =value(rhs(%))+c;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%$IntG6$*$)-%#ln G6#%\"xG\"\"#\"\"\"F+" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/%!G,&*&)-%#l nG6#%\"xG\"\"#\"\"\"F+F-F--%$IntG6$,$*&F,F-F(F-F-F+!\"\"" }}{PARA 11 " " 1 "" {XPPMATH 20 "6#/%!G,**&)-%#lnG6#%\"xG\"\"#\"\"\"F+F-F-*(F,F-F(F -F+F-!\"\"*&F,F-F+F-F-%\"cGF-" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {PARA 0 "" 0 "" {TEXT -1 46 "More details are shown by using the proce dure " }{TEXT 0 10 "intbyparts" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 75 "Int(ln(x)^2, x);\n``=intbyparts(%,u=ln(x)^2,v,info=true);\n``=value(rhs(%))+c;" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#-%$IntG6$*$)-%#lnG6#%\"xG\"\"#\"\"\"F+ " }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%*PIECEWISEG6$7$/%\"uG*$)-%#lnG6# %\"xG\"\"#\"\"\"/%\"vGF.7$/**%\"dGF0F(F0%#~dG!\"\"F.F8,$*(F/F0F+F0F.F8 F0/**F6F0F2F0F7F8F.F8F0" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%$IntG6$* $)-%#lnG6#%\"xG\"\"#\"\"\"F,-F%6$*&%\"uGF.-%!G6#**%\"dGF.%\"vGF.%#~dG! \"\"F,F:F.F," }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/%!G,&*&%\"uG\"\"\"%\" vGF(F(-%$IntG6$*&F)F(-F$6#**%\"dGF(F'F(%#~dG!\"\"%\"xGF3F(F4F3" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#/%!G,&*&)-%#lnG6#%\"xG\"\"#\"\"\"F+F-F --%$IntG6$,$*&F,F-F(F-F-F+!\"\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/%! G,**&)-%#lnG6#%\"xG\"\"#\"\"\"F+F-F-*(F,F-F(F-F+F-!\"\"*&F,F-F+F-F-%\" cGF-" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 10 "Example 5 \+ " }}{PARA 258 "" 0 "" {TEXT 286 8 "Question" }{TEXT -1 2 ": " }}{PARA 0 "" 0 "" {TEXT -1 37 "(a) Use integration by parts to find " } {XPPEDIT 18 0 "Int(arctan*x,x);" "6#-%$IntG6$*&%'arctanG\"\"\"%\"xGF(F )" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 54 "(b) Find the area o f the region bounded by the graph " }{XPPEDIT 18 0 "y = arctan*x;" "6 #/%\"yG*&%'arctanG\"\"\"%\"xGF'" }{TEXT -1 6 ", the " }{TEXT 284 1 "x " }{TEXT -1 19 " axis and the line " }{XPPEDIT 18 0 "x = sqrt(3);" "6# /%\"xG-%%sqrtG6#\"\"$" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 54 " (c) Find the area of the region bounded by the graph " }{XPPEDIT 18 0 "y = tan*x;" "6#/%\"yG*&%$tanG\"\"\"%\"xGF'" }{TEXT -1 6 ", the " } {TEXT 285 1 "y" }{TEXT -1 19 " axis and the line " }{XPPEDIT 18 0 "y = sqrt(3);" "6#/%\"yG-%%sqrtG6#\"\"$" }{TEXT -1 2 ". " }}{PARA 0 "" 0 " " {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 287 8 "Solution" }{TEXT -1 2 " : " }}{PARA 0 "" 0 "" {TEXT -1 4 "(a) " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "Int(arctan*x,x);" "6#-%$IntG6$*&%'arctanG\"\"\"% \"xGF(F)" }{TEXT -1 10 " ... " }{XPPEDIT 18 0 "PIECEWISE([u = arc tan*x, v = x],[du/dx = 1/(1+x^2), dv/dx = 1]);" "6#-%*PIECEWISEG6$7$/% \"uG*&%'arctanG\"\"\"%\"xGF+/%\"vGF,7$/*&%#duGF+%#dxG!\"\"*&F+F+,&F+F+ *$F,\"\"#F+F4/*&%#dvGF+F3F4F+" }{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 1 " " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "``= \+ Int(u*``(dv/dx),x)" "6#/%!G-%$IntG6$*&%\"uG\"\"\"-F$6#*&%#dvGF*%#dxG! \"\"F*%\"xG" }{TEXT -1 1 " " }}{PARA 256 "" 0 "" {TEXT -1 1 " " } {XPPEDIT 18 0 "`` = u*v-Int(v*``(du/dx),x)" "6#/%!G,&*&%\"uG\"\"\"%\"v GF(F(-%$IntG6$*&F)F(-F$6#*&%#duGF(%#dxG!\"\"F(%\"xGF3" }{TEXT -1 1 " \+ " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "`` = x*arctan*x-I nt(x*`.`*``(1/(1+x^2)),x);" "6#/%!G,&*(%\"xG\"\"\"%'arctanGF(F'F(F(-%$ IntG6$*(F'F(%\".GF(-F$6#*&F(F(,&F(F(*$F'\"\"#F(!\"\"F(F'F5" }{TEXT -1 1 " " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "`` = x*arctan *x-Int(x/(1+x^2),x);" "6#/%!G,&*(%\"xG\"\"\"%'arctanGF(F'F(F(-%$IntG6$ *&F'F(,&F(F(*$F'\"\"#F(!\"\"F'F1" }{TEXT -1 1 " " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "`` = x*arctan*x-1/2;" "6#/%!G,&*(%\"xG \"\"\"%'arctanGF(F'F(F(*&F(F(\"\"#!\"\"F," }{TEXT -1 1 " " }{XPPEDIT 18 0 "Int(2*x/(1+x^2),x)" "6#-%$IntG6$*(\"\"#\"\"\"%\"xGF(,&F(F(*$F)F' F(!\"\"F)" }{TEXT -1 1 " " }}{PARA 256 "" 0 "" {TEXT -1 1 " " } {XPPEDIT 18 0 "`` = x*arctan*x-1/2;" "6#/%!G,&*(%\"xG\"\"\"%'arctanGF( F'F(F(*&F(F(\"\"#!\"\"F," }{TEXT -1 1 " " }{XPPEDIT 18 0 "ln(1+x^2) +c " "6#,&-%#lnG6#,&\"\"\"F(*$%\"xG\"\"#F(F(%\"cGF(" }{TEXT -1 1 "," }} {PARA 0 "" 0 "" {TEXT -1 19 "since the integral " }{XPPEDIT 18 0 "Int( 2*x/(1+x^2),x)" "6#-%$IntG6$*(\"\"#\"\"\"%\"xGF(,&F(F(*$F)F'F(!\"\"F) " }{TEXT -1 14 " has the form " }{XPPEDIT 18 0 "Int(`f '`(x)/f(x),x); " "6#-%$IntG6$*&-%$f~'G6#%\"xG\"\"\"-%\"fG6#F*!\"\"F*" }{TEXT -1 6 " w ith " }{XPPEDIT 18 0 "f(x) = 1+x^2;" "6#/-%\"fG6#%\"xG,&\"\"\"F)*$F'\" \"#F)" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 56 "student[intparts](Int(arctan(x),x), arctan(x));\nvalue(%);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,&*&-%'arcta nG6#%\"xG\"\"\"F(F)F)-%$IntG6$*&F(F),&F)F)*$)F(\"\"#F)F)!\"\"F(F2" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#,&*&-%'arctanG6#%\"xG\"\"\"F(F)F)*&#F) \"\"#F)-%#lnG6#,&F)F)*$)F(F,F)F)F)!\"\"" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 46 "More details are shown by using th e procedure " }{TEXT 0 10 "intbyparts" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 79 "Int(arc tan(x),x);\n``=intbyparts(%,u=arctan(x),v,info=true);\n``=value(rhs(%) )+c;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%$IntG6$-%'arctanG6#%\"xGF)" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%*PIECEWISEG6$7$/%\"uG-%'arctanG6#% \"xG/%\"vGF,7$/**%\"dG\"\"\"F(F3%#~dG!\"\"F,F5*&F3F3,&F3F3*$)F,\"\"#F3 F3F5/**F2F3F.F3F4F5F,F5F3" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%$IntG6 $-%'arctanG6#%\"xGF*-F%6$*&%\"uG\"\"\"-%!G6#**%\"dGF/%\"vGF/%#~dG!\"\" F*F7F/F*" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/%!G,&*&%\"uG\"\"\"%\"vGF( F(-%$IntG6$*&F)F(-F$6#**%\"dGF(F'F(%#~dG!\"\"%\"xGF3F(F4F3" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/%!G,&*&%\"xG\"\"\"-%'arctanG6#F'F(F(-%$IntG 6$*&F'F(,&F(F(*$)F'\"\"#F(F(!\"\"F'F4" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/%!G,(*&%\"xG\"\"\"-%'arctanG6#F'F(F(*&#F(\"\"#F(-%#lnG6#,&F(F(*$)F 'F.F(F(F(!\"\"%\"cGF(" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 4 "(b) " 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i]m7$$\"3kh4q2;\\)e\"FV$\"3%yy()4_Z*35FV7$Fa^mF)7&Fe^mF`^m7$$\"3d^RmW# 3Gi\"FV$\"3DOgeF(Q&=5FV7$Fh^mF)7&F\\_mFg^m7$$\"3!p%H;PpRf;FV$\"3\"RL$4 (HY%G5FV7$F__mF)7&Fc_mF^_m7$$\"3/_k\\%)pT%p\"FV$\"3](Q,x_Lw.\"FV7$Ff_m F)7&Fj_mFe_m7$$\"3*)*****z!30KZ5FV7$F]`mF)7\"-%&STYLEG6# %,PATCHNOGRIDG-%&COLORG6&Fiz$\"#&)!\"#Fj`mFj`m-F$6%7$7$F($\"3c'*[zEjzq :FV7$FbzFaam-Fgz6&FizF)F)F)-%*LINESTYLEG6#\"\"$-F$6$7$7$F]`mF(7$F]`m$ \"3k(f'>^v>Z5FVFdam-%%TEXTG6%7$$\"#b!\"\"$FgbmFgbmQ\"x6\"Fdam-Fbbm6%7$ Fhbm$\"#>FgbmQ\"yFjbmFdam-Fbbm6&7$$!#:F\\am$\"$d\"F\\amQ$p/2FjbmFdam-% %FONTG6$%'SYMBOLG\"\"*-%*AXESTICKSG6$7$/\"\"#%\"2G/\"\"%%\"4G7$/F)%\"0 G/\"\"\"%\"1G-%+AXESLABELSG6%%!GFaem-Fjcm6#%(DEFAULTG-%%VIEWG6$;FdcmFe bm;FhbmF^cm" 1 2 0 1 10 0 2 9 1 4 2 1.000000 46.000000 45.000000 0 0 " Curve 1" "Curve 2" "Curve 3" "Curve 4" "Curve 5" "Curve 6" "Curve 7" } }{TEXT -1 1 " " }}{PARA 256 "" 0 "" {TEXT -1 2 " " }}{PARA 258 "" 0 " " {TEXT -1 22 "The required area is: " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "Int(arctan*x,x = 0 .. sqrt(3));" "6#-%$IntG6$*&%'a rctanG\"\"\"%\"xGF(/F);\"\"!-%%sqrtG6#\"\"$" }{TEXT -1 1 " " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "`` = x*arctan*x-1/2;" "6#/% !G,&*(%\"xG\"\"\"%'arctanGF(F'F(F(*&F(F(\"\"#!\"\"F," }{TEXT -1 1 " " }{XPPEDIT 18 0 "ln(1+x^2);" "6#-%#lnG6#,&\"\"\"F'*$%\"xG\"\"#F'" } {TEXT -1 1 " " }{XPPEDIT 18 0 "PIECEWISE([sqrt(3),``],[0,``])" "6#-%*P IECEWISEG6$7$-%%sqrtG6#\"\"$%!G7$\"\"!F+" }{TEXT -1 1 " " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "`` = sqrt(3)*arctan(sqrt(3))-1/ 2;" "6#/%!G,&*&-%%sqrtG6#\"\"$\"\"\"-%'arctanG6#-F(6#F*F+F+*&F+F+\"\"# !\"\"F3" }{TEXT -1 1 " " }{XPPEDIT 18 0 "ln*4-0;" "6#,&*&%#lnG\"\"\"\" \"%F&F&\"\"!!\"\"" }{TEXT -1 1 " " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "`` = sqrt(3)*Pi/3-ln*2;" "6#/%!G,&*(-%%sqrtG6#\"\"$\" \"\"%#PiGF+F*!\"\"F+*&%#lnGF+\"\"#F+F-" }{TEXT -1 1 " " }{TEXT 288 1 " ~" }{TEXT -1 14 " 1.120652184. 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}}}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 9 "Example 6" }}{PARA 258 "" 0 "" {TEXT 275 8 "Question" }{TEXT -1 9 ": Find " }{XPPEDIT 18 0 "Int(exp(x)*cos*x, x);" "6#-%$IntG6$*(-%$expG6#%\"xG\"\"\"%$cosGF+F*F+F*" }{TEXT -1 6 " \+ and " }{XPPEDIT 18 0 "Int(exp(x)*sin*x,x);" "6#-%$IntG6$*(-%$expG6#%\" xG\"\"\"%$sinGF+F*F+F*" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT 276 8 "Solution" }{TEXT -1 2 ": " }}{PARA 0 "" 0 "" {TEXT -1 28 "Consider \+ the first integral " }{XPPEDIT 18 0 "Int(exp(x)*cos*x,x);" "6#-%$IntG6 $*(-%$expG6#%\"xG\"\"\"%$cosGF+F*F+F*" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 4 "Let " }{XPPEDIT 18 0 "u = exp(x);" "6#/%\"uG-%$expG6# %\"xG" }{TEXT -1 5 " and " }{XPPEDIT 18 0 "v = sin*x;" "6#/%\"vG*&%$si nG\"\"\"%\"xGF'" }{TEXT -1 38 " in the integration by parts formula. \+ " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "Int(exp(x)*cos*x, x);" "6#-%$IntG6$*(-%$expG6#%\"xG\"\"\"%$cosGF+F*F+F*" }{TEXT -1 7 " . .. " }{XPPEDIT 18 0 "PIECEWISE([u = exp(x), v = sin*x],[du/dx = exp( x), dv/dx = cos*x]);" "6#-%*PIECEWISEG6$7$/%\"uG-%$expG6#%\"xG/%\"vG*& %$sinG\"\"\"F,F17$/*&%#duGF1%#dxG!\"\"-F*6#F,/*&%#dvGF1F6F7*&%$cosGF1F ,F1" }{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 258 "" 0 "" {TEXT -1 6 "Hence " }{XPPEDIT 18 0 "Int(u*``(dv/dx),x) = u*v-Int(v* ``(du/dx),x)" "6#/-%$IntG6$*&%\"uG\"\"\"-%!G6#*&%#dvGF)%#dxG!\"\"F)%\" xG,&*&F(F)%\"vGF)F)-F%6$*&F4F)-F+6#*&%#duGF)F/F0F)F1F0" }{TEXT -1 11 " becomes: " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "Int(ex p(x)*cos*x,x) = exp(x)*sin*x-Int(exp(x)*sin*x,x);" "6#/-%$IntG6$*(-%$e xpG6#%\"xG\"\"\"%$cosGF,F+F,F+,&*(-F)6#F+F,%$sinGF,F+F,F,-F%6$*(-F)6#F +F,F2F,F+F,F+!\"\"" }{TEXT -1 14 " ------- (i). " }}{PARA 0 "" 0 "" {TEXT -1 37 "The integral that has been obtained, " }{XPPEDIT 18 0 "In t(exp(x)*sin*x,x);" "6#-%$IntG6$*(-%$expG6#%\"xG\"\"\"%$sinGF+F*F+F*" }{TEXT -1 101 ", is no simpler than the original integral. It is, in f act, the other integral which we need to find." }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "Int(exp(x)*sin*x,x);" "6#-%$IntG6$*(-%$ expG6#%\"xG\"\"\"%$sinGF+F*F+F*" }{TEXT -1 7 " ... " }{XPPEDIT 18 0 "PIECEWISE([u = exp(x), v = -cos*x],[du/dx = exp(x), dv/dx = sin*x]); " "6#-%*PIECEWISEG6$7$/%\"uG-%$expG6#%\"xG/%\"vG,$*&%$cosG\"\"\"F,F2! \"\"7$/*&%#duGF2%#dxGF3-F*6#F,/*&%#dvGF2F8F3*&%$sinGF2F,F2" }{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 258 "" 0 "" {TEXT -1 6 " Hence " }{XPPEDIT 18 0 "Int(u*``(dv/dx),x) = u*v-Int(v*``(du/dx),x)" " 6#/-%$IntG6$*&%\"uG\"\"\"-%!G6#*&%#dvGF)%#dxG!\"\"F)%\"xG,&*&F(F)%\"vG F)F)-F%6$*&F4F)-F+6#*&%#duGF)F/F0F)F1F0" }{TEXT -1 11 " becomes: " }} {PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "Int(exp(x)*sin*x,x) = -exp(x)*cos*x-Int(exp(x)*(-cos*x),x);" "6#/-%$IntG6$*(-%$expG6#%\"xG \"\"\"%$sinGF,F+F,F+,&*(-F)6#F+F,%$cosGF,F+F,!\"\"-F%6$*&-F)6#F+F,,$*& F2F,F+F,F3F,F+F3" }{TEXT -1 2 ", " }}{PARA 258 "" 0 "" {TEXT -1 9 "tha t is, " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "Int(exp(x)* sin*x,x) = -exp(x)*cos*x+Int(exp(x)*cos*x,x);" "6#/-%$IntG6$*(-%$expG6 #%\"xG\"\"\"%$sinGF,F+F,F+,&*(-F)6#F+F,%$cosGF,F+F,!\"\"-F%6$*(-F)6#F+ F,F2F,F+F,F+F," }{TEXT -1 14 " ------- (ii)." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 63 "The equations (i) and (ii) can \+ be solved for the two integrals " }{XPPEDIT 18 0 "Int(exp(x)*cos*x,x); " "6#-%$IntG6$*(-%$expG6#%\"xG\"\"\"%$cosGF+F*F+F*" }{TEXT -1 5 " and \+ " }{XPPEDIT 18 0 "Int(exp(x)*sin*x,x);" "6#-%$IntG6$*(-%$expG6#%\"xG\" \"\"%$sinGF+F*F+F*" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 45 "Equ ation (ii) can be rewritten in the form: " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "Int(exp(x)*cos*x,x) = exp(x)*cos*x+Int(exp(x) *sin*x,x);" "6#/-%$IntG6$*(-%$expG6#%\"xG\"\"\"%$cosGF,F+F,F+,&*(-F)6# F+F,F-F,F+F,F,-F%6$*(-F)6#F+F,%$sinGF,F+F,F+F," }{TEXT -1 15 " ------- (iii)." }}{PARA 0 "" 0 "" {TEXT -1 38 "Adding equations (i) and (iii) gives: " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "2*Int(exp (x)*cos*x,x) = exp(x)*sin*x+exp(x)*cos*x;" "6#/*&\"\"#\"\"\"-%$IntG6$* (-%$expG6#%\"xGF&%$cosGF&F.F&F.F&,&*(-F,6#F.F&%$sinGF&F.F&F&*(-F,6#F.F &F/F&F.F&F&" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 60 "Hence, af ter adding a constant of integration, we see that " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "Int(exp(x)*cos*x,x) = (exp(x)*sin*x+ exp(x)*cos*x)/2+c;" "6#/-%$IntG6$*(-%$expG6#%\"xG\"\"\"%$cosGF,F+F,F+, &*&,&*(-F)6#F+F,%$sinGF,F+F,F,*(-F)6#F+F,F-F,F+F,F,F,\"\"#!\"\"F,%\"cG F," }{TEXT -1 1 " " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "``=1/2" "6#/%!G*&\"\"\"F&\"\"#!\"\"" }{TEXT -1 1 " " }{XPPEDIT 18 0 " exp(x)*(sin*x+cos*x)+c;" "6#,&*&-%$expG6#%\"xG\"\"\",&*&%$sinGF)F(F)F) *&%$cosGF)F(F)F)F)F)%\"cGF)" }{TEXT -1 3 ". " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{TEXT 278 14 "______________" }{TEXT -1 1 " " }}{PARA 258 "" 0 "" {TEXT -1 0 "" }}{PARA 258 "" 0 "" {TEXT -1 43 "Equation (i ) can be rewritten in the form: " }}{PARA 256 "" 0 "" {TEXT -1 1 " " } {XPPEDIT 18 0 "Int(exp(x)*sin*x,x) = exp(x)*sin*x-Int(exp(x)*cos*x,x); " "6#/-%$IntG6$*(-%$expG6#%\"xG\"\"\"%$sinGF,F+F,F+,&*(-F)6#F+F,F-F,F+ F,F,-F%6$*(-F)6#F+F,%$cosGF,F+F,F+!\"\"" }{TEXT -1 14 " ------- (iv). " }}{PARA 0 "" 0 "" {TEXT -1 38 "Adding equations (ii) and (iv) gives: " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "2*Int(exp(x)*sin *x,x) = exp(x)*sin*x-exp(x)*cos*x;" "6#/*&\"\"#\"\"\"-%$IntG6$*(-%$exp G6#%\"xGF&%$sinGF&F.F&F.F&,&*(-F,6#F.F&F/F&F.F&F&*(-F,6#F.F&%$cosGF&F. F&!\"\"" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 60 "Hence, after \+ adding a constant of integration, we see that " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "Int(exp(x)*sin*x,x) = (exp(x)*sin*x-exp (x)*cos*x)/2+c;" "6#/-%$IntG6$*(-%$expG6#%\"xG\"\"\"%$sinGF,F+F,F+,&*& ,&*(-F)6#F+F,F-F,F+F,F,*(-F)6#F+F,%$cosGF,F+F,!\"\"F,\"\"#F8F,%\"cGF, " }{TEXT -1 1 " " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "` `=1/2" "6#/%!G*&\"\"\"F&\"\"#!\"\"" }{TEXT -1 1 " " }{XPPEDIT 18 0 "ex p(x)*(sin*x-cos*x)+c;" "6#,&*&-%$expG6#%\"xG\"\"\",&*&%$sinGF)F(F)F)*& %$cosGF)F(F)!\"\"F)F)%\"cGF)" }{TEXT -1 2 ". " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{TEXT 279 14 "______________" }{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 36 "In t(exp(x)*cos(x),x);\n``=value(%)+c;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6 #-%$IntG6$*&-%$expG6#%\"xG\"\"\"-%$cosGF)F+F*" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/%!G,(*&#\"\"\"\"\"#F(*&-%$expG6#%\"xGF(-%$cosGF-F(F(F( *&F'F(*&F+F(-%$sinGF-F(F(F(%\"cGF(" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 36 "Int(exp(x)*sin(x),x);\n``=value(%)+c;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%$IntG6$*&-%$expG6#%\"xG\"\"\"-%$sinGF)F+F*" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#/%!G,(*&#\"\"\"\"\"#F(*&-%$expG6#%\"xG F(-%$cosGF-F(F(!\"\"*&#F(F)F(*&F+F(-%$sinGF-F(F(F(%\"cGF(" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 101 "More details of \+ the application of the integration by parts formula are shown by using the procedure " }{TEXT 0 10 "intbyparts" }{TEXT -1 2 ". " }}{PARA 0 " " 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 60 "Int(e xp(x)*cos(x),x);\n``=intbyparts(%,u=exp(x),v,info=true);" }}{PARA 11 " " 1 "" {XPPMATH 20 "6#-%$IntG6$*&-%$expG6#%\"xG\"\"\"-%$cosGF)F+F*" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#-%*PIECEWISEG6$7$/%\"uG-%$expG6#%\"xG/ %\"vG-%$sinGF+7$/**%\"dG\"\"\"F(F5%#~dG!\"\"F,F7F)/**F4F5F.F5F6F7F,F7- %$cosGF+" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%$IntG6$*&-%$expG6#%\"xG \"\"\"-%$cosGF*F,F+-F%6$*&%\"uGF,-%!G6#**%\"dGF,%\"vGF,%#~dG!\"\"F+F:F ,F+" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/%!G,&*&%\"uG\"\"\"%\"vGF(F(-%$ IntG6$*&F)F(-F$6#**%\"dGF(F'F(%#~dG!\"\"%\"xGF3F(F4F3" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/%!G,&*&-%$expG6#%\"xG\"\"\"-%$sinGF)F+F+-%$IntG6$ F&F*!\"\"" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{PARA 0 " " 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 81 "Int(e xp(x)*sin(x),x);\n``=intbyparts(%,u=exp(x),v,info=true);\n``=simplify( rhs(%));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%$IntG6$*&-%$expG6#%\"xG \"\"\"-%$sinGF)F+F*" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%*PIECEWISEG6$ 7$/%\"uG-%$expG6#%\"xG/%\"vG,$-%$cosGF+!\"\"7$/**%\"dG\"\"\"F(F7%#~dGF 2F,F2F)/**F6F7F.F7F8F2F,F2-%$sinGF+" }}{PARA 11 "" 1 "" {XPPMATH 20 "6 #/-%$IntG6$*&-%$expG6#%\"xG\"\"\"-%$sinGF*F,F+-F%6$*&%\"uGF,-%!G6#**% \"dGF,%\"vGF,%#~dG!\"\"F+F:F,F+" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/%! G,&*&%\"uG\"\"\"%\"vGF(F(-%$IntG6$*&F)F(-F$6#**%\"dGF(F'F(%#~dG!\"\"% \"xGF3F(F4F3" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/%!G,&*&-%$expG6#%\"xG \"\"\"-%$cosGF)F+!\"\"-%$IntG6$,$F&F.F*F." }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/%!G,&*&-%$expG6#%\"xG\"\"\"-%$cosGF)F+!\"\"-%$IntG6$F& F*F+" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 10 "Example 7 \+ " }}{PARA 258 "" 0 "" {TEXT 295 8 "Question" }{TEXT -1 13 ": Find (a ) " }{XPPEDIT 18 0 "Int(arctan(sqrt(x)),x);" "6#-%$IntG6$-%'arctanG6#- %%sqrtG6#%\"xGF," }{TEXT -1 12 " and (b) " }{XPPEDIT 18 0 "Int(x*ar ctan(sqrt(x)),x);" "6#-%$IntG6$*&%\"xG\"\"\"-%'arctanG6#-%%sqrtG6#F'F( F'" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT 296 8 "Solution" }{TEXT -1 3 ": " }}{PARA 0 "" 0 "" {TEXT -1 70 "The integration by parts for mula can be used to find these integrals. " }}{PARA 0 "" 0 "" {TEXT -1 3 "(a)" }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "Int(arct an(sqrt(x)),x);" "6#-%$IntG6$-%'arctanG6#-%%sqrtG6#%\"xGF," }{TEXT -1 10 " ... " }{XPPEDIT 18 0 "PIECEWISE([u = arctan(sqrt(x)), v = x] ,[du/dx = 1/((1+x)*2*sqrt(x)), dv/dx = 1]);" "6#-%*PIECEWISEG6$7$/%\"u G-%'arctanG6#-%%sqrtG6#%\"xG/%\"vGF/7$/*&%#duG\"\"\"%#dxG!\"\"*&F6F6*( ,&F6F6F/F6F6\"\"#F6-F-6#F/F6F8/*&%#dvGF6F7F8F6" }{TEXT -1 1 " " }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 256 "" 0 "" {TEXT -1 1 " " } {XPPEDIT 18 0 "``= Int(u*``(dv/dx),x)" "6#/%!G-%$IntG6$*&%\"uG\"\"\"-F $6#*&%#dvGF*%#dxG!\"\"F*%\"xG" }{TEXT -1 1 " " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "`` = u*v-Int(v*``(du/dx),x)" "6#/%!G,&* &%\"uG\"\"\"%\"vGF(F(-%$IntG6$*&F)F(-F$6#*&%#duGF(%#dxG!\"\"F(%\"xGF3 " }{TEXT -1 1 " " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "` ` = x*arctan(sqrt(x))-Int(x*`.`*``(1/(2*sqrt(x)*(1+x))),x);" "6#/%!G,& *&%\"xG\"\"\"-%'arctanG6#-%%sqrtG6#F'F(F(-%$IntG6$*(F'F(%\".GF(-F$6#*& F(F(*(\"\"#F(-F-6#F'F(,&F(F(F'F(F(!\"\"F(F'F<" }{TEXT -1 1 " " }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 256 "" 0 "" {TEXT -1 1 " " } {XPPEDIT 18 0 "``=x*arctan(sqrt(x))-Int(sqrt(x)/(2*(1+x)),x)" "6#/%!G, &*&%\"xG\"\"\"-%'arctanG6#-%%sqrtG6#F'F(F(-%$IntG6$*&-F-6#F'F(*&\"\"#F (,&F(F(F'F(F(!\"\"F'F8" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 13 "The integral " }{XPPEDIT 18 0 "Int(sqrt(x)/(2*(1+x)),x);" "6#-%$In tG6$*&-%%sqrtG6#%\"xG\"\"\"*&\"\"#F+,&F+F+F*F+F+!\"\"F*" }{TEXT -1 41 " can be found by making the substitution " }{XPPEDIT 18 0 "z=sqrt(x) " "6#/%\"zG-%%sqrtG6#%\"xG" }{TEXT -1 26 ", or inverse substitution " }{XPPEDIT 18 0 "x=z^2" "6#/%\"xG*$%\"zG\"\"#" }{TEXT -1 2 ". " }} {PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "Int(sqrt(x)/(2*(1+x)) ,x);" "6#-%$IntG6$*&-%%sqrtG6#%\"xG\"\"\"*&\"\"#F+,&F+F+F*F+F+!\"\"F* " }{TEXT -1 8 " ... " }{XPPEDIT 18 0 "PIECEWISE([z = sqrt(x), x = z ^2],[dz = ``(1/(2*sqrt(x)))*`.`*dx, dx = 2*z*dz]);" "6#-%*PIECEWISEG6$ 7$/%\"zG-%%sqrtG6#%\"xG/F,*$F(\"\"#7$/%#dzG*(-%!G6#*&\"\"\"F8*&F/F8-F* 6#F,F8!\"\"F8%\".GF8%#dxGF8/F>*(F/F8F(F8F2F8" }{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "``=Int(z^2/(1+z^2),z)" "6#/%!G-%$IntG6$*&%\"zG\"\"#,&\"\"\"F,*$F )F*F,!\"\"F)" }{TEXT -1 1 " " }{XPPEDIT 18 0 "`` = Int(1-1/(1+z^2),z); " "6#/%!G-%$IntG6$,&\"\"\"F)*&F)F),&F)F)*$%\"zG\"\"#F)!\"\"F/F-" } {TEXT -1 1 " " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "`` = z-arctan*z+c;" "6#/%!G,(%\"zG\"\"\"*&%'arctanGF'F&F'!\"\"%\"cGF'" } {TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "`` = sqrt(x)-arctan(sqrt(x))+c;" "6#/%! G,(-%%sqrtG6#%\"xG\"\"\"-%'arctanG6#-F'6#F)!\"\"%\"cGF*" }{TEXT -1 2 " . " }}{PARA 0 "" 0 "" {TEXT -1 16 "It follows that " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "Int(arctan(sqrt(x)),x) = x*arctan(sq rt(x))-sqrt(x)+arctan(sqrt(x))+c[1];" "6#/-%$IntG6$-%'arctanG6#-%%sqrt G6#%\"xGF-,**&F-\"\"\"-F(6#-F+6#F-F0F0-F+6#F-!\"\"-F(6#-F+6#F-F0&%\"cG 6#F0F0" }{TEXT -1 2 ", " }}{PARA 0 "" 0 "" {TEXT -1 6 "where " } {XPPEDIT 18 0 "c[1]=-c" "6#/&%\"cG6#\"\"\",$F%!\"\"" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 258 "" 0 "" {TEXT -1 4 "Let " }{XPPEDIT 18 0 "f(x)=x*arctan(sqrt(x))-sqrt(x)+arctan(sqrt(x))" "6#/-% \"fG6#%\"xG,(*&F'\"\"\"-%'arctanG6#-%%sqrtG6#F'F*F*-F/6#F'!\"\"-F,6#-F /6#F'F*" }{TEXT -1 6 " and " }{XPPEDIT 18 0 "g(x) = arctan(sqrt(x)); " "6#/-%\"gG6#%\"xG-%'arctanG6#-%%sqrtG6#F'" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 39 "In the followin g picture the graph of " }{XPPEDIT 18 0 "f(x)" "6#-%\"fG6#%\"xG" } {TEXT -1 13 " is drawn in " }{TEXT 260 3 "red" }{TEXT -1 20 " while th e graph of " }{XPPEDIT 18 0 "g(x);" "6#-%\"gG6#%\"xG" }{TEXT -1 13 " i s drawn in " }{TEXT 256 4 "blue" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 45 "The picture is consistent with the fact that " }{XPPEDIT 18 0 "Diff([f(x)],x) = g(x);" "6#/-%%DiffG6$7#-%\"fG6#%\"xGF+-%\"gG6#F +" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 " > " 0 "" {MPLTEXT 1 0 148 "f := x -> x*arctan(sqrt(x))-sqrt(x)+arctan( sqrt(x)):\n'f(x)'=f(x);\ng := x -> arctan(sqrt(x)):\n'g(x)'=g(x);\nplo t([f(x),g(x)],x=0..4,color=[red,blue]);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%\"fG6#%\"xG,(*&F'\"\"\"-%'arctanG6#*$F'#F*\"\"#F*F*F.!\"\"F+F *" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%\"gG6#%\"xG-%'arctanG6#*$F'#\" \"\"\"\"#" }}{PARA 13 "" 1 "" {GLPLOT2D 346 305 305 {PLOTDATA 2 "6&-%' CURVESG6$7S7$$\"\"!F)F(7$$\"3Hmmmm;')=()!#>$\"31ZyG+*euo\"F-7$$\"3RLLL e'40j\"!#=$\"3i&pM2XI`D%F-7$$\"3mmmm;6m$[#F3$\"3c\\b1IKI!)yF-7$$\"3fmm m;yYULF3$\"3^h'p#G)*f77F37$$\"3%HLL$eF>(>%F3$\"3-VfV^5$Ho\"F37$$\"3Qmm m\">K'*)\\F3$\"3Whgfx\"\\Z:#F37$$\"3P*****\\Kd,\"eF3$\"3-y(>@LyXn#F37$ $\"3-mmm\"fX(emF3$\"3*f*zM[$)fTKF37$$\"3.*****\\U7Y](F3$\"3;H%GZ%\\@LQ F37$$\"3'QLLLV!pu$)F3$\"35)HzYS'RmWF37$$\"3xmmm;c0T\"*F3$\"3gDe]xm%G/& F37$$\"3#*******H,Q+5!#<$\"3WD\")e#)*[4r&F37$$\"3)*******\\*3q3\"F`o$ \"3*=u(\\,(G0S'F37$$\"3)*******p=\\q6F`o$\"3Uq#eMW993(F37$$\"3mmm;fBIY 7F`o$\"3Ug*GP5>Er(F37$$\"3GLLLj$[kL\"F`o$\"3yH2[5r/y%)F37$$\"3?LLL`Q\" GT\"F`o$\"3\\.%emU&GQ\"*F37$$\"3!*****\\s]k,:F`o$\"3#fce2?e!>**F37$$\" 39LLL`dF!e\"F`o$\"3uT](3a%4i5F`o7$$\"33++]sgam;F`o$\"3l#>]7r(>S6F`o7$$ \"3/++]F`o$\"3EH'Rv\"R>p8F`o7$$\"3immmTc-)*>F`o$\"3W'*)fWQ])\\ 9F`o7$$\"3Mmm;f`@'3#F`o$\"3_&G'Hm+`M:F`o7$$\"3y****\\nZ)H;#F`o$\"3B=N# H&f&*3;F`o7$$\"3YmmmJy*eC#F`o$\"3WFL&)y)[+p\"F`o7$$\"3')******R^bJBF`o $\"3'*4[rVxcuF`o7$$\"3k*****\\@fke#F`o$\"3R8n+)*QBI?F`o7$$\" 3/LLL`4NnEF`o$\"39Y&p./zD6#F`o7$$\"3#*******\\,s`FF`o$\"34Ih-_E5,AF`o7 $$\"3[mm;zM)>$GF`o$\"3K+(yz/O=G#F`o7$$\"3$*******pfak qBF`o7$$\"3#HLLeg`!)*HF`o$\"3g2Mvu7qaCF`o7$$\"3w****\\#G2A3$F`o$\"3ElH \"pxnIa#F`o7$$\"3;LLL$)G[kJF`o$\"32U\"f)Q+%*HEF`o7$$\"3#)****\\7yh]KF` o$\"3yHn'))fw8s#F`o7$$\"3xmmm')fdLLF`o$\"3,d$zet)*)4GF`o7$$\"3bmmm,FT= MF`o$\"3-(Q7V9v3!HF`o7$$\"3FLL$e#pa-NF`o$\"3G\"eGuPL:*HF`o7$$\"3!***** **Rv&)zNF`o$\"3#G-Z-`4_2$F`o7$$\"3ILLLGUYoOF`o$\"33P%z#pa`rJF`o7$$\"3_ mmm1^rZPF`o$\"3]&*G1EH1eKF`o7$$\"34++]sI@KQF`o$\"33,,a6ap]LF`o7$$\"34+ +]2%)38RF`o$\"3!\\a=2<5(RMF`o7$$\"\"%F)$\"3/_/(*)eVd`$F`o-%'COLOURG6&% $RGBG$\"*++++\"!\")F(F(-F$6$7YF'7$$\"3ILLL3x&)*3\"F-$\"3M?RVVX>S5F37$$ \"3emmm;arz@F-$\"3S?'=*pqzl9F37$$\"3')*****\\7t&pKF-$\"3oZ4mBr'))y\"F3 7$$\"39LLLL3VfVF-$\"3c**36gTNe?F37$$\"3s******\\i9RlF-$\"3vHRw**f_.DF3 7$F+$\"3s$*fRbK=rGF37$$\"3-++]7z>^7F3$\"35Fk5CX()*R$F37$F1$\"3p4F[=CuP QF37$F7$\"3Z5pp()\\PBYF37$F<$\"3*>ER$)o9>C&F37$FA$\"3#o;'y&fv'[dF37$FF $\"3?nNuFa!*\\hF37$FK$\"3=*y**G]7H^'F37$FP$\"3E!y:T3!GWoF37$FU$\"3gL/r g]wQrF37$FZ$\"35*oDQJM6T(F37$Fin$\"3+n+\\,S`HwF37$F^o$\"3K.5Ry<$\\&yF3 7$Fdo$\"3E]8Std\\i!)F37$Fio$\"3Q\"fz&Gg8Z#)F37$F^p$\"3seW!3sDLS)F37$Fc p$\"3[S1/HI\\w&)F37$Fhp$\"3\"y`PPksOr)F37$F]q$\"3!eqp2,cM'))F37$Fbq$\" 3?H!3XXF\"))*)F37$Fgq$\"3O\"Gc`v2s6*F37$F\\r$\"3:`AyQ:KL#*F37$Far$\"39 R>R2g/[$*F37$Ffr$\"3#[(p4M+6[%*F37$F[s$\"33AG'o.Q3b*F37$F`s$\"3Q%[^IQ( G_'*F37$Fes$\"3Z.m![Iylt*F37$Fjs$\"3th\\w,`sB)*F37$F_t$\"3B2XG9*=)4**F 37$Fdt$\"3;TA*fC\\/***F37$Fit$\"3j&R)H27`15F`o7$F^u$\"3s!y*>&40X,\"F`o 7$Fcu$\"3\"H^EY@'Q@5F`o7$Fhu$\"3k`%pl3c%G5F`o7$F]v$\"3!GJ.h?IY.\"F`o7$ Fbv$\"3\"RGLvrU6/\"F`o7$Fgv$\"3-R7JDq0Z5F`o7$F\\w$\"3chC%GlII0\"F`o7$F aw$\"3%pM+JJw'e5F`o7$Ffw$\"3)Qrmi*>Rk5F`o7$F[x$\"3+!35#['=(p5F`o7$F`x$ \"3w#\\/zn%*\\2\"F`o7$Fex$\"3+*euO%[1!3\"F`o7$Fjx$\"3q2$p(o%)e%3\"F`o7 $F_y$\"3ox**3,:i*3\"F`o7$Fdy$\"3o4]$\\Y#*R4\"F`o7$Fiy$\"3u%oR!RM_)4\"F `o7$F^z$\"3HZt$4ySF5\"F`o7$Fcz$\"3T!4%z<([r5\"F`o-Fhz6&FjzF(F(F[[l-%+A XESLABELSG6$Q\"x6\"Q!Fefl-%%VIEWG6$;F(Fcz%(DEFAULTG" 1 2 0 1 10 0 2 9 1 4 2 1.000000 45.000000 45.000000 0 0 "Curve 1" "Curve 2" }}}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 142 "I nt(1/2*sqrt(x)/(1+x),x);\n``=student[changevar](sqrt(x)=z,%);\n``=map( convert,rhs(%),parfrac,z);\n``=value(rhs(%))+c;\n``=subs(z=sqrt(x),rhs (%));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%$IntG6$,$*(\"\"#!\"\"%\"xG# \"\"\"F(,&F,F,F*F,F)F,F*" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/%!G-%$Int G6$*&%\"zG\"\"#,&\"\"\"F,*$)F)F*F,F,!\"\"F)" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/%!G-%$IntG6$,&\"\"\"F)*&F)F),&F)F)*$)%\"zG\"\"#F)F)!\" \"F0F." }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/%!G,(%\"zG\"\"\"-%'arctanG6 #F&!\"\"%\"cGF'" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/%!G,(*$%\"xG#\"\" \"\"\"#F)-%'arctanG6#F&!\"\"%\"cGF)" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 84 "Int(arctan(sqrt(x)),x);\n` `=student[intparts](%,arctan(sqrt(x)));\n``=value(rhs(%))+c;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%$IntG6$-%'arctanG6#*$%\"xG#\"\"\"\"\"#F*" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/%!G,&*&-%'arctanG6#*$%\"xG#\"\"\"\" \"#F-F+F-F--%$IntG6$,$*(F.!\"\"F+F,,&F-F-F+F-F4F-F+F4" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/%!G,**&-%'arctanG6#*$%\"xG#\"\"\"\"\"#F-F+F-F-F*! \"\"F'F-%\"cGF-" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 46 "More details are shown by using the procedure " }{TEXT 0 10 "intbyparts" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 91 "Int(arctan(sqrt(x)),x);\n``= intbyparts(%,u=arctan(sqrt(x)),v,info=true);\n``=value(rhs(%))+c;" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#-%$IntG6$-%'arctanG6#*$%\"xG#\"\"\"\" \"#F*" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%*PIECEWISEG6$7$/%\"uG-%'arc tanG6#*$%\"xG#\"\"\"\"\"#/%\"vGF-7$/**%\"dGF/F(F/%#~dG!\"\"F-F8,$*&F/F /*(F0F/F-#F/F0,&F-F/F/F/F/F8F//**F6F/F2F/F7F8F-F8F/" }}{PARA 11 "" 1 " " {XPPMATH 20 "6#/-%$IntG6$-%'arctanG6#*$%\"xG#\"\"\"\"\"#F+-F%6$*&%\" uGF--%!G6#**%\"dGF-%\"vGF-%#~dG!\"\"F+F:F-F+" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/%!G,&*&%\"uG\"\"\"%\"vGF(F(-%$IntG6$*&F)F(-F$6#**%\"dG F(F'F(%#~dG!\"\"%\"xGF3F(F4F3" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/%!G, &*&-%'arctanG6#*$%\"xG#\"\"\"\"\"#F-F+F-F--%$IntG6$,$*(F.!\"\"F+F,,&F+ F-F-F-F4F-F+F4" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/%!G,**&-%'arctanG6# *$%\"xG#\"\"\"\"\"#F-F+F-F-F*!\"\"F'F-%\"cGF-" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 3 "(b)" }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "Int(x*arctan(sqrt(x)),x);" "6#-%$IntG6$*&%\"xG\"\"\"-%'arctanG6#-%% sqrtG6#F'F(F'" }{TEXT -1 10 " ... " }{XPPEDIT 18 0 "PIECEWISE([u \+ = arctan(sqrt(x)), v = x^2/2],[du/dx = 1/((1+x)*2*sqrt(x)), dv/dx = x] );" "6#-%*PIECEWISEG6$7$/%\"uG-%'arctanG6#-%%sqrtG6#%\"xG/%\"vG*&F/\" \"#F3!\"\"7$/*&%#duG\"\"\"%#dxGF4*&F9F9*(,&F9F9F/F9F9F3F9-F-6#F/F9F4/* &%#dvGF9F:F4F/" }{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "``= Int(u*``(dv/dx),x )" "6#/%!G-%$IntG6$*&%\"uG\"\"\"-F$6#*&%#dvGF*%#dxG!\"\"F*%\"xG" } {TEXT -1 1 " " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "`` = u*v-Int(v*``(du/dx),x)" "6#/%!G,&*&%\"uG\"\"\"%\"vGF(F(-%$IntG6$*&F)F (-F$6#*&%#duGF(%#dxG!\"\"F(%\"xGF3" }{TEXT -1 1 " " }}{PARA 256 "" 0 " " {TEXT -1 1 " " }{XPPEDIT 18 0 "`` = ``(x^2/2)*arctan(sqrt(x))-Int(`` (x^2/2)*`.`*``(1/(2*sqrt(x)*(1+x))),x);" "6#/%!G,&*&-F$6#*&%\"xG\"\"#F +!\"\"\"\"\"-%'arctanG6#-%%sqrtG6#F*F-F--%$IntG6$*(-F$6#*&F*F+F+F,F-% \".GF--F$6#*&F-F-*(F+F--F26#F*F-,&F-F-F*F-F-F,F-F*F," }{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 256 "" 0 "" {TEXT -1 1 " " } {XPPEDIT 18 0 "``=1/2" "6#/%!G*&\"\"\"F&\"\"#!\"\"" }{TEXT -1 1 " " } {XPPEDIT 18 0 "x^2*arctan(sqrt(x))-1/2;" "6#,&*&%\"xG\"\"#-%'arctanG6# -%%sqrtG6#F%\"\"\"F-*&F-F-F&!\"\"F/" }{TEXT -1 1 " " }{XPPEDIT 18 0 "I nt(x*sqrt(x)/(2*(1+x)),x)" "6#-%$IntG6$*(%\"xG\"\"\"-%%sqrtG6#F'F(*&\" \"#F(,&F(F(F'F(F(!\"\"F'" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 13 "The integral " }{XPPEDIT 18 0 "Int(x*sqrt(x)/(2*(1+x)),x);" "6#-%$ IntG6$*(%\"xG\"\"\"-%%sqrtG6#F'F(*&\"\"#F(,&F(F(F'F(F(!\"\"F'" }{TEXT -1 41 " can be found by making the substitution " }{XPPEDIT 18 0 "z=sq rt(x)" "6#/%\"zG-%%sqrtG6#%\"xG" }{TEXT -1 2 ". " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "Int(x*sqrt(x)/(2*(1+x)),x);" "6#-%$IntG 6$*(%\"xG\"\"\"-%%sqrtG6#F'F(*&\"\"#F(,&F(F(F'F(F(!\"\"F'" }{TEXT -1 8 " ... " }{XPPEDIT 18 0 "PIECEWISE([z=sqrt(x),z^2=x],[dz=``(1/(2*s qrt(x)))*`.`*dx,2*z*dz=dx])" "6#-%*PIECEWISEG6$7$/%\"zG-%%sqrtG6#%\"xG /*$F(\"\"#F,7$/%#dzG*(-%!G6#*&\"\"\"F8*&F/F8-F*6#F,F8!\"\"F8%\".GF8%#d xGF8/*(F/F8F(F8F2F8F>" }{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 0 " " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "`` = Int(z^4/(1+z ^2),z);" "6#/%!G-%$IntG6$*&%\"zG\"\"%,&\"\"\"F,*$F)\"\"#F,!\"\"F)" } {TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 256 "" 0 "" {TEXT -1 43 " " }{XPPEDIT 18 0 "z^2 +0*z-1" "6#,(*$%\"zG\"\"#\"\"\"*&\"\"!F'F%F'F'F'!\"\"" } {TEXT -1 1 " " }}{PARA 256 "" 0 "" {TEXT -1 41 " __ ___________________" }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "z^2+0*z+1" "6#,(*$%\"zG\"\"#\"\"\"*&\"\"!F'F%F'F'F'F'" }{TEXT -1 5 " | " }{XPPEDIT 18 0 "z^4+0*z^3+0*z^2+0*z+0" "6#,,*$%\"zG\"\"%\"\"\" *&\"\"!F'*$F%\"\"$F'F'*&F)F'*$F%\"\"#F'F'*&F)F'F%F'F'F)F'" }{TEXT -1 1 " " }}{PARA 256 "" 0 "" {TEXT -1 9 " " }{XPPEDIT 18 0 "z^4+0 *z^3+1*z^2;" "6#,(*$%\"zG\"\"%\"\"\"*&\"\"!F'*$F%\"\"$F'F'*&F'F'*$F%\" \"#F'F'" }{TEXT -1 1 " " }}{PARA 256 "" 0 "" {TEXT -1 19 " ______ ______ " }}{PARA 256 "" 0 "" {TEXT -1 40 " \+ " }{XPPEDIT 18 0 "-z^2+0*z+0" "6#,(*$%\"zG\"\"#!\"\"*&\"\" !\"\"\"F%F*F*F)F*" }{TEXT -1 1 " " }}{PARA 256 "" 0 "" {TEXT -1 40 " \+ " }{XPPEDIT 18 0 "-z^2+0*z-1;" " 6#,(*$%\"zG\"\"#!\"\"*&\"\"!\"\"\"F%F*F*F*F'" }{TEXT -1 1 " " }}{PARA 256 "" 0 "" {TEXT -1 52 " ____ ______ " }}{PARA 256 "" 0 "" {TEXT -1 61 " \+ 1 " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {PARA 0 "" 0 "" {TEXT -1 26 "This division shows that " }{XPPEDIT 18 0 "z^4/(1+z^2) = z^2-1+1/(1+z^2);" "6#/*&%\"zG\"\"%,&\"\"\"F(*$F%\"\"# F(!\"\",(*$F%F*F(F(F+*&F(F(,&F(F(*$F%F*F(F+F(" }{TEXT -1 2 ". " }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 15 "Alternati vely, " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "(z^2-1)*(z^ 2+1)=z^4-1" "6#/*&,&*$%\"zG\"\"#\"\"\"F)!\"\"F),&*$F'F(F)F)F)F),&*$F' \"\"%F)F)F*" }{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 3 "so " }} {PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "z^2-1 = z^4/(1+z^2)-1 /(1+z^2);" "6#/,&*$%\"zG\"\"#\"\"\"F(!\"\",&*&F&\"\"%,&F(F(*$F&F'F(F)F (*&F(F(,&F(F(*$F&F'F(F)F)" }{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 12 "which gives " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "z ^4/(1+z^2) = z^2-1+1/(1+z^2);" "6#/*&%\"zG\"\"%,&\"\"\"F(*$F%\"\"#F(! \"\",(*$F%F*F(F(F+*&F(F(,&F(F(*$F%F*F(F+F(" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 6 "Hence " }}{PARA 256 "" 0 "" {TEXT -1 1 " " } {XPPEDIT 18 0 "Int(z^4/(1+z^2),z)=Int(z^2-1+1/(1+z^2),z)" "6#/-%$IntG6 $*&%\"zG\"\"%,&\"\"\"F+*$F(\"\"#F+!\"\"F(-F%6$,(*$F(F-F+F+F.*&F+F+,&F+ F+*$F(F-F+F.F+F(" }{XPPEDIT 18 0 "``=z^3/3-z+arctan*z+c" "6#/%!G,**&% \"zG\"\"$F(!\"\"\"\"\"F'F)*&%'arctanGF*F'F*F*%\"cGF*" }{TEXT -1 2 ", \+ " }}{PARA 0 "" 0 "" {TEXT -1 8 "so that " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "Int(x*sqrt(x)/(2*(1+x)),x)=1/3" "6#/-%$IntG6$ *(%\"xG\"\"\"-%%sqrtG6#F(F)*&\"\"#F),&F)F)F(F)F)!\"\"F(*&F)F)\"\"$F0" }{TEXT -1 1 " " }{XPPEDIT 18 0 "x^(3/2)-sqrt(x)+arctan(sqrt(x))+c" "6# ,*)%\"xG*&\"\"$\"\"\"\"\"#!\"\"F(-%%sqrtG6#F%F*-%'arctanG6#-F,6#F%F(% \"cGF(" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 1 " " }}{PARA 0 " " 0 "" {TEXT -1 16 "It follows that " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "Int(x*arctan(sqrt(x)),x) =1/2" "6#/-%$IntG6$*&%\"xG \"\"\"-%'arctanG6#-%%sqrtG6#F(F)F(*&F)F)\"\"#!\"\"" }{TEXT -1 1 " " } {XPPEDIT 18 0 "x^2*arctan(sqrt(x))-1/6;" "6#,&*&%\"xG\"\"#-%'arctanG6# -%%sqrtG6#F%\"\"\"F-*&F-F-\"\"'!\"\"F0" }{TEXT -1 1 " " }{XPPEDIT 18 0 "x^(3/2)+1/2;" "6#,&)%\"xG*&\"\"$\"\"\"\"\"#!\"\"F(*&F(F(F)F*F(" } {TEXT -1 1 " " }{XPPEDIT 18 0 "sqrt(x)-1/2;" "6#,&-%%sqrtG6#%\"xG\"\" \"*&F(F(\"\"#!\"\"F+" }{TEXT -1 1 " " }{XPPEDIT 18 0 "arctan(sqrt(x))+ c[1]" "6#,&-%'arctanG6#-%%sqrtG6#%\"xG\"\"\"&%\"cG6#F+F+" }{TEXT -1 2 ", " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 6 "whe re " }{XPPEDIT 18 0 "c[1] = -c/2;" "6#/&%\"cG6#\"\"\",$*&F%F'\"\"#!\" \"F+" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 39 "In the following picture the graph of " }{XPPEDIT 18 0 "f(x)" "6#-%\"fG6#%\"xG" }{TEXT -1 13 " is drawn in " }{TEXT 260 3 " red" }{TEXT -1 20 " while the graph of " }{XPPEDIT 18 0 "g(x);" "6#-% \"gG6#%\"xG" }{TEXT -1 13 " is drawn in " }{TEXT 256 4 "blue" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 45 "The picture is consistent wit h the fact that " }{XPPEDIT 18 0 "Diff([f(x)],x) = g(x);" "6#/-%%DiffG 6$7#-%\"fG6#%\"xGF+-%\"gG6#F+" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 168 "f := x -> x ^2/2*arctan(sqrt(x))-x^(3/2)/6+sqrt(x)/2-arctan(sqrt(x))/2:\n'f(x)'=f( x);\ng := x -> x*arctan(sqrt(x)):\n'g(x)'=g(x);\nplot([f(x),g(x)],x=0. .5,color=[red,blue]);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%\"fG6#%\"x G,**&#\"\"\"\"\"#F+*&)F'F,F+-%'arctanG6#*$F'F*F+F+F+*&\"\"'!\"\"F'#\" \"$F,F5*&F,F5F'F*F+*&#F+F,F+F/F+F5" }}{PARA 11 "" 1 "" {XPPMATH 20 "6# /-%\"gG6#%\"xG*&F'\"\"\"-%'arctanG6#*$F'#F)\"\"#F)" }}{PARA 13 "" 1 " " {GLPLOT2D 285 350 350 {PLOTDATA 2 "6&-%'CURVESG6$7S7$$\"\"!F)F(7$$\" 3GLLL3x&)*3\"!#=$\"3vV+K\")euH:!#?7$$\"3umm\"H2P\"Q?F-$\"33`kFr1QorF07 $$\"3MLL$eRwX5$F-$\"3W^Ae2$**)3?!#>7$$\"33ML$3x%3yTF-$\"3.%GUw%>7NTF;7 $$\"3emm\"z%4\\Y_F-$\"3E3dj'*y$o;(F;7$$\"3`LLeR-/PiF-$\"3'=)GE/ln&3\"F -7$$\"3]***\\il'pisF-$\"3/;9?QY*>c\"F-7$$\"3>MLe*)>VB$)F-$\"39s6!=&\\+ g@F-7$$\"3Y++DJbw!Q*F-$\"3i))\\df!Qn'GF-7$$\"3%ommTIOo/\"!#<$\"3hG.5^M e7PF-7$$\"3YLL3_>jU6Fgn$\"3#R--[9S+c%F-7$$\"37++]i^Z]7Fgn$\"3%[\">!o)[ VJcF-7$$\"33++](=h(e8Fgn$\"3>ds/4XrMoF-7$$\"3/++]P[6j9Fgn$\"3N[!Rh@zs6 )F-7$$\"3UL$e*[z(yb\"Fgn$\"3+&H,V&)*4)Q*F-7$$\"3wmm;a/cq;Fgn$\"3iedD)H ZK5\"Fgn7$$\"3%ommmJFgn$\"3sztT!pJ>i\"Fgn7$$\"3K+]i!f#=$3#Fgn$ \"3pG=XF%Q<$=Fgn7$$\"3?+](=xpe=#Fgn$\"3g8?f/8[W?Fgn7$$\"37nm\"H28IH#Fg n$\"3Qn1Ujw.!G#Fgn7$$\"3um;zpSS\"R#Fgn$\"33PB10al3DFgn7$$\"3GLL3_?`(\\ #Fgn$\"3w(*)f74W&oFFgn7$$\"3fL$e*)>pxg#Fgn$\"3)[UT-?&>`IFgn7$$\"33+]Pf 4t.FFgn$\"3E)p\"fk3D8LFgn7$$\"3uLLe*Gst!GFgn$\"3&fWL^Y6qg$Fgn7$$\"30++ +DRW9HFgn$\"3**))ft8siCRFgn7$$\"3:++DJE>>IFgn$\"3Osu*[W=$\\UFgn7$$\"3F +]i!RU07$Fgn$\"37^oOl>owXFgn7$$\"3+++v=S2LKFgn$\"3k0\"***\\#\\a&\\Fgn7 $$\"3Jmmm\"p)=MLFgn$\"3Az.!o#[c4`Fgn7$$\"3B++](=]@W$Fgn$\"3\"*>vII06-d Fgn7$$\"35L$e*[$z*RNFgn$\"3]&p%*fN^22'Fgn7$$\"3e++]iC$pk$Fgn$\"3%=-l4e &*y['Fgn7$$\"3[m;H2qcZPFgn$\"3wZ04h(4R*oFgn7$$\"3O+]7.\"fF&QFgn$\"35S9 S'HiBL(Fgn7$$\"3Ymm;/OgbRFgn$\"3=@8y2h&\\x(Fgn7$$\"3w**\\ilAFjSFgn$\"3 ?*f[hD2JD)Fgn7$$\"3yLLL$)*pp;%Fgn$\"3!*)\\\")[G#*Fgn7$$\"3Cn;HdO=yVFgn$\"33-Qqn\\))Q(*Fgn7$$\"3a+++D> #[Z%Fgn$\"3s&H=je-@-\"!#;7$$\"3SnmT&G!e&e%Fgn$\"3A([DDN!))y5F_y7$$\"3# RLLL)Qk%o%Fgn$\"35#H&>91/J6F_y7$$\"37+]iSjE!z%Fgn$\"3A*4JHJ!4)=\"F_y7$ $\"3a+]P40O\"*[Fgn$\"3/z`&43&3W7F_y7$$\"\"&F)$\"3%\\7j0zydI\"F_y-%'COL OURG6&%$RGBG$\"*++++\"!\")F(F(-F$6$7SF'7$F+$\"35E)e$)zr^Z$F;7$F2$\"3xk _BI%HIk)F;7$F7$\"3k9&RtN%>y:F-7$F=$\"3O0a?7w\\(R#F-7$FB$\"3Y;6*oT]))G$ F-7$FG$\"3;$H'34q?pTF-7$FL$\"3M\\*GK\"R&e7&F-7$FQ$\"3KAdj^P)e:'F-7$FV$ \"3H%Ge&z\"[x@(F-7$Fen$\"3Q>$oji8;M)F-7$F[o$\"3%R9U&4(3[N*F-7$F`o$\"3G #Ri(eN&=0\"Fgn7$Feo$\"3];d9,J!4<\"Fgn7$Fjo$\"39$***[;()\\(G\"Fgn7$F_p$ \"3gOQa\"3>[R\"Fgn7$Fdp$\"3_%pl;VdS_\"Fgn7$Fip$\"3n/0\"e&\\![j\"Fgn7$F ^q$\"3C0\\!y$)p\\w\"Fgn7$Fcq$\"3_4kcD$)H\")=Fgn7$Fhq$\"3a')=*)[%Q+,#Fg n7$F]r$\"3W6Bc_DjL@Fgn7$Fbr$\"3!)z]Jy3cjAFgn7$Fgr$\"3QRimXCp$Q#Fgn7$F \\s$\"37e]\"[6#49DFgn7$Fas$\"3UM#)*)R$)Q]EFgn7$Ffs$\"3u&4y$)e'ppFFgn7$ F[t$\"3S27?0g?**GFgn7$F`t$\"36pc;\"3wO.$Fgn7$Fet$\"3m8My$Rce;$Fgn7$Fjt $\"3'=\\^<23VH$Fgn7$F_u$\"3Er%zL\"[aPMFgn7$Fdu$\"3cc]C+HxmNFgn7$Fiu$\" 3Jl'4e6s_q$Fgn7$F^v$\"3Y`cz)f;7$QFgn7$Fcv$\"3mA>e2%o$pRFgn7$Fhv$\"3')H d:u)z(*4%Fgn7$F]w$\"3j2\\'>!e^OUFgn7$Fbw$\"3M8w!*oofqVFgn7$Fgw$\"329g- K1P6XFgn7$F\\x$\"3!fW#GPsKZYFgn7$Fax$\"39P&RZ5Gny%Fgn7$Ffx$\"3&p]W57E` #\\Fgn7$F[y$\"3WL:;M3)H0&Fgn7$Fay$\"3-!*G*\\bD'*>&Fgn7$Ffy$\"3G*psEy\" 3J`Fgn7$F[z$\"3)*yux(GQ:Z&Fgn7$F`z$\"3 \+ " 0 "" {MPLTEXT 1 0 144 "Int(1/2*x*sqrt(x)/(1+x),x);\n``=student[chang evar](sqrt(x)=z,%);\n``=map(convert,rhs(%),parfrac,z);\n``=value(rhs(% ))+c;\n``=subs(z=sqrt(x),rhs(%));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#- %$IntG6$,$*(\"\"#!\"\"%\"xG#\"\"$F(,&\"\"\"F.F*F.F)F.F*" }}{PARA 11 " " 1 "" {XPPMATH 20 "6#/%!G-%$IntG6$*&%\"zG\"\"%,&\"\"\"F,*$)F)\"\"#F,F ,!\"\"F)" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/%!G-%$IntG6$,(*$)%\"zG\" \"#\"\"\"F-F-!\"\"*&F-F-,&F-F-F)F-F.F-F+" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/%!G,**&\"\"$!\"\"%\"zGF'\"\"\"F)F(-%'arctanG6#F)F*%\"cGF*" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#/%!G,**&\"\"$!\"\"%\"xG#F'\"\"#\"\"\"* $F)#F,F+F(-%'arctanG6#F-F,%\"cGF," }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 86 "Int(x*arctan(sqrt(x)),x);\n` `=student[intparts](%,arctan(sqrt(x)));\n``=value(rhs(%))+c;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%$IntG6$*&-%'arctanG6#*$%\"xG#\"\"\"\"\"#F- F+F-F+" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/%!G,&*&#\"\"\"\"\"#F(*&-%'a rctanG6#*$%\"xGF'F()F/F)F(F(F(-%$IntG6$,$*(\"\"%!\"\"F/#\"\"$F),&F(F(F /F(F7F(F/F7" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/%!G,,*&#\"\"\"\"\"#F(* &-%'arctanG6#*$%\"xGF'F()F/F)F(F(F(*&\"\"'!\"\"F/#\"\"$F)F3*&F)F3F/F'F (*&#F(F)F(F+F(F3%\"cGF(" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 " " 0 "" {TEXT -1 46 "More details are shown by using the procedure " } {TEXT 0 10 "intbyparts" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 93 "Int(x*arctan(sqrt(x)),x );\n``=intbyparts(%,u=arctan(sqrt(x)),v,info=true);\n``=value(rhs(%))+ c;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%$IntG6$*&-%'arctanG6#*$%\"xG# \"\"\"\"\"#F-F+F-F+" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%*PIECEWISEG6$ 7$/%\"uG-%'arctanG6#*$%\"xG#\"\"\"\"\"#/%\"vG,$*&F0!\"\"F-F0F/7$/**%\" dGF/F(F/%#~dGF5F-F5,$*&F/F/*(F0F/F-#F/F0,&F-F/F/F/F/F5F//**F9F/F2F/F:F 5F-F5F-" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%$IntG6$*&-%'arctanG6#*$% \"xG#\"\"\"\"\"#F.F,F.F,-F%6$*&%\"uGF.-%!G6#**%\"dGF.%\"vGF.%#~dG!\"\" F,F;F.F," }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/%!G,&*&%\"uG\"\"\"%\"vGF( F(-%$IntG6$*&F)F(-F$6#**%\"dGF(F'F(%#~dG!\"\"%\"xGF3F(F4F3" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/%!G,&*&#\"\"\"\"\"#F(*&-%'arctanG6#*$%\"xGF 'F()F/F)F(F(F(-%$IntG6$,$*(\"\"%!\"\"F/#\"\"$F),&F/F(F(F(F7F(F/F7" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#/%!G,,*&#\"\"\"\"\"#F(*&-%'arctanG6#*$ %\"xGF'F()F/F)F(F(F(*&\"\"'!\"\"F/#\"\"$F)F3*&F)F3F/F'F(*&#F(F)F(F+F(F 3%\"cGF(" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 10 "Example 8 \+ " }}{PARA 258 "" 0 "" {TEXT 291 8 "Question" }{TEXT -1 2 ": " }}{PARA 0 "" 0 "" {TEXT -1 9 "Find (a) " }{XPPEDIT 18 0 "Int(sin*sqrt(x),x);" "6#-%$IntG6$*&%$sinG\"\"\"-%%sqrtG6#%\"xGF(F," }{TEXT -1 6 " (b) " } {XPPEDIT 18 0 "Int(sin(sqrt(x)),x=0..Pi^2/4)" "6#-%$IntG6$-%$sinG6#-%% sqrtG6#%\"xG/F,;\"\"!*&%#PiG\"\"#\"\"%!\"\"" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT 292 8 "Solution" }{TEXT -1 2 ": " }}{PARA 0 "" 0 "" {TEXT -1 72 "(a) The integration by parts formula can be used to find \+ this integral. " }}{PARA 0 "" 0 "" {TEXT -1 16 "First note that " }} {PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "d/dx" "6#*&%\"dG\"\" \"%#dxG!\"\"" }{TEXT -1 1 " " }{XPPEDIT 18 0 "[-cos(sqrt(x))] = sin(sq rt(x))/(2*sqrt(x))" "6#/7#,$-%$cosG6#-%%sqrtG6#%\"xG!\"\"*&-%$sinG6#-F *6#F,\"\"\"*&\"\"#F4-F*6#F,F4F-" }{TEXT -1 15 " ------- (i). " }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 256 "" 0 "" {TEXT -1 1 " " } {XPPEDIT 18 0 "Int(sin*sqrt(x),x) = Int(2*sqrt(x)*`.`*``(sin(sqrt(x))/ (2*sqrt(x))),x)" "6#/-%$IntG6$*&%$sinG\"\"\"-%%sqrtG6#%\"xGF)F--F%6$** \"\"#F)-F+6#F-F)%\".GF)-%!G6#*&-F(6#-F+6#F-F)*&F1F)-F+6#F-F)!\"\"F)F- " }{TEXT -1 8 " ... " }{XPPEDIT 18 0 "PIECEWISE([u = 2*sqrt(x), v = -cos(sqrt(x))],[du/dx = 1/sqrt(x), dv/dx = sin(sqrt(x))/(2*sqrt(x))]) " "6#-%*PIECEWISEG6$7$/%\"uG*&\"\"#\"\"\"-%%sqrtG6#%\"xGF+/%\"vG,$-%$c osG6#-F-6#F/!\"\"7$/*&%#duGF+%#dxGF8*&F+F+-F-6#F/F8/*&%#dvGF+F=F8*&-%$ sinG6#-F-6#F/F+*&F*F+-F-6#F/F+F8" }{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "``= I nt(u*``(dv/dx),x)" "6#/%!G-%$IntG6$*&%\"uG\"\"\"-F$6#*&%#dvGF*%#dxG!\" \"F*%\"xG" }{TEXT -1 1 " " }}{PARA 256 "" 0 "" {TEXT -1 1 " " } {XPPEDIT 18 0 "`` = u*v-Int(v*``(du/dx),x)" "6#/%!G,&*&%\"uG\"\"\"%\"v GF(F(-%$IntG6$*&F)F(-F$6#*&%#duGF(%#dxG!\"\"F(%\"xGF3" }{TEXT -1 1 " \+ " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "`` = 2*sqrt(x)*`. `*``(-cos*sqrt(x))-Int(``(-cos(sqrt(x)))*`.`*``(1/sqrt(x)),x);" "6#/%! G,&**\"\"#\"\"\"-%%sqrtG6#%\"xGF(%\".GF(-F$6#,$*&%$cosGF(-F*6#F,F(!\" \"F(F(-%$IntG6$*(-F$6#,$-F26#-F*6#F,F5F(F-F(-F$6#*&F(F(-F*6#F,F5F(F,F5 " }{TEXT -1 1 " " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "` `=-2*sqrt(x)*cos*sqrt(x)+Int(cos*sqrt(x)/sqrt(x),x)" "6#/%!G,&**\"\"# \"\"\"-%%sqrtG6#%\"xGF(%$cosGF(-F*6#F,F(!\"\"-%$IntG6$*(F-F(-F*6#F,F(- F*6#F,F0F,F(" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {PARA 0 "" 0 "" {TEXT -1 37 "The companion result to (i) above is " }} {PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "d/dx" "6#*&%\"dG\"\" \"%#dxG!\"\"" }{TEXT -1 1 " " }{XPPEDIT 18 0 "[sin*sqrt(x)] = cos*sqrt (x)/(2*sqrt(x))" "6#/7#*&%$sinG\"\"\"-%%sqrtG6#%\"xGF'*(%$cosGF'-F)6#F +F'*&\"\"#F'-F)6#F+F'!\"\"" }{TEXT -1 16 " ------- (ii). " }}{PARA 0 "" 0 "" {TEXT -1 6 "Hence " }{XPPEDIT 18 0 "Int(cos*sqrt(x)/sqrt(x),x) = 2*sin*sqrt(x)+c;" "6#/-%$IntG6$*(%$cosG\"\"\"-%%sqrtG6#%\"xGF)-F+6# F-!\"\"F-,&*(\"\"#F)%$sinGF)-F+6#F-F)F)%\"cGF)" }{TEXT -1 2 ". " }} {PARA 0 "" 0 "" {TEXT -1 15 "Alternatively, " }{XPPEDIT 18 0 "Int(cos* sqrt(x)/sqrt(x),x)" "6#-%$IntG6$*(%$cosG\"\"\"-%%sqrtG6#%\"xGF(-F*6#F, !\"\"F," }{TEXT -1 41 " can be found by making the substitution " } {XPPEDIT 18 0 "z = sqrt(x);" "6#/%\"zG-%%sqrtG6#%\"xG" }{TEXT -1 2 ". \+ " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "Int(cos*sqrt(x)/s qrt(x),x)" "6#-%$IntG6$*(%$cosG\"\"\"-%%sqrtG6#%\"xGF(-F*6#F,!\"\"F," }{TEXT -1 8 " ... " }{XPPEDIT 18 0 "PIECEWISE([z = sqrt(x), ``],[dz = 1/(2*sqrt(x)), 2*`.`*dz = 1/sqrt(x)]);" "6#-%*PIECEWISEG6$7$/%\"zG- %%sqrtG6#%\"xG%!G7$/%#dzG*&\"\"\"F2*&\"\"#F2-F*6#F,F2!\"\"/*(F4F2%\".G F2F0F2*&F2F2-F*6#F,F7" }{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 0 " " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "``=2*Int(cos*z,z) " "6#/%!G*&\"\"#\"\"\"-%$IntG6$*&%$cosGF'%\"zGF'F-F'" }{XPPEDIT 18 0 " ``=2*sin*z+c[1]" "6#/%!G,&*(\"\"#\"\"\"%$sinGF(%\"zGF(F(&%\"cG6#F(F(" }{XPPEDIT 18 0 "`` = 2*sin*sqrt(x)+c;" "6#/%!G,&*(\"\"#\"\"\"%$sinGF(- %%sqrtG6#%\"xGF(F(%\"cGF(" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 16 "It follows that " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "Int(sin*sqrt(x),x)=-2*sqrt(x)*cos*sqrt(x)+2*sin*sqrt(x)+c" "6#/- %$IntG6$*&%$sinG\"\"\"-%%sqrtG6#%\"xGF)F-,(**\"\"#F)-F+6#F-F)%$cosGF)- F+6#F-F)!\"\"*(F0F)F(F)-F+6#F-F)F)%\"cGF)" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 64 "stud ent[intparts](Int(sin(sqrt(x)),x),2*sqrt(x));\n``=value(%)+c;" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#,&*(\"\"#\"\"\"-%$cosG6#*$%\"xG#F&F%F& F+F,!\"\"-%$IntG6$,$*&F'F&F+#F-F%F-F+F-" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/%!G,(*&\"\"#\"\"\"-%$sinG6#*$%\"xG#F(F'F(F(*(F'F(-%$cosGF+F(F-F .!\"\"%\"cGF(" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 46 "More details are shown by using the procedure " }{TEXT 0 10 "intbyparts" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 82 "Int(sin(sqrt(x)),x);\n``=int byparts(%,u=2*sqrt(x),v,info=true);\n``=value(rhs(%))+c;" }}{PARA 11 " " 1 "" {XPPMATH 20 "6#-%$IntG6$-%$sinG6#*$%\"xG#\"\"\"\"\"#F*" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#-%*PIECEWISEG6$7$/%\"uG,$*&\"\"#\"\"\" %\"xG#F,F+F,/%\"vG,$-%$cosG6#*$F-F.!\"\"7$/**%\"dGF,F(F,%#~dGF6F-F6*&F ,F,*$F-#F,F+F6/**F:F,F0F,F;F6F-F6,$*&F.F,*&-%$sinGF4F,F-#F6F+F,F," }} {PARA 11 "" 1 "" {XPPMATH 20 "6#/-%$IntG6$-%$sinG6#*$%\"xG#\"\"\"\"\"# F+-F%6$*&%\"uGF--%!G6#**%\"dGF-%\"vGF-%#~dG!\"\"F+F:F-F+" }}{PARA 11 " " 1 "" {XPPMATH 20 "6#/%!G,&*&%\"uG\"\"\"%\"vGF(F(-%$IntG6$*&F)F(-F$6# **%\"dGF(F'F(%#~dG!\"\"%\"xGF3F(F4F3" }}{PARA 11 "" 1 "" {XPPMATH 20 " 6#/%!G,&*(\"\"#\"\"\"%\"xG#F(F'-%$cosG6#*$F)F*F(!\"\"-%$IntG6$,$*&F+F( F)#F/F'F/F)F/" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/%!G,(*(\"\"#\"\"\"% \"xG#F(F'-%$cosG6#*$F)F*F(!\"\"*&F'F(-%$sinGF-F(F(%\"cGF(" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 4 "(b) " }}{PARA 256 "" 0 "" {TEXT -1 1 " \+ " }{XPPEDIT 18 0 "Int(sin(sqrt(x)),x = 0 .. Pi^2/4)=-2*sqrt(x)*cos*sqr t(x)+2*sin*sqrt(x)" "6#/-%$IntG6$-%$sinG6#-%%sqrtG6#%\"xG/F-;\"\"!*&%# PiG\"\"#\"\"%!\"\",&**F3\"\"\"-F+6#F-F8%$cosGF8-F+6#F-F8F5*(F3F8F(F8-F +6#F-F8F8" }{TEXT -1 1 " " }{XPPEDIT 18 0 "PIECEWISE([Pi^2/4,``],[0,`` ])" "6#-%*PIECEWISEG6$7$*&%#PiG\"\"#\"\"%!\"\"%!G7$\"\"!F," }{TEXT -1 1 " " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "``=0-2*sin(Pi /2)" "6#/%!G,&\"\"!\"\"\"*&\"\"#F'-%$sinG6#*&%#PiGF'F)!\"\"F'F/" } {XPPEDIT 18 0 " ``=2" "6#/%!G\"\"#" }{TEXT -1 2 ". 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" }}{PARA 0 "" 0 "" {TEXT 299 8 "Solution" }{TEXT -1 2 ": " }} {PARA 0 "" 0 "" {TEXT -1 48 "Applying the integration by parts formula with " }{XPPEDIT 18 0 "u = (arcsin*x)^2;" "6#/%\"uG*$*&%'arcsinG\"\" \"%\"xGF(\"\"#" }{TEXT -1 5 " and " }{XPPEDIT 18 0 "v= x" "6#/%\"vG%\" xG" }{TEXT -1 8 " gives: " }}{PARA 256 "" 0 "" {TEXT -1 1 " " } {XPPEDIT 18 0 "Int((arcsin*x)^2,x)" "6#-%$IntG6$*$*&%'arcsinG\"\"\"%\" xGF)\"\"#F*" }{TEXT -1 8 " ... " }{XPPEDIT 18 0 "PIECEWISE([u = (ar csin*x)^2, v = x],[du/dx = 2*arcsin*x/sqrt(1-x^2), dv/dx = 1])" "6#-%* PIECEWISEG6$7$/%\"uG*$*&%'arcsinG\"\"\"%\"xGF,\"\"#/%\"vGF-7$/*&%#duGF ,%#dxG!\"\"**F.F,F+F,F-F,-%%sqrtG6#,&F,F,*$F-F.F6F6/*&%#dvGF,F5F6F," } {TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "``= Int(u*``(dv/dx),x)" "6#/%!G-%$IntG6 $*&%\"uG\"\"\"-F$6#*&%#dvGF*%#dxG!\"\"F*%\"xG" }{TEXT -1 1 " " }} {PARA 256 "" 0 "" {TEXT -1 2 " " }{XPPEDIT 18 0 "`` = u*v-Int(v*``(du /dx),x)" "6#/%!G,&*&%\"uG\"\"\"%\"vGF(F(-%$IntG6$*&F)F(-F$6#*&%#duGF(% #dxG!\"\"F(%\"xGF3" }{TEXT -1 1 " " }}{PARA 256 "" 0 "" {TEXT -1 1 " \+ " }{XPPEDIT 18 0 "``=x*(arcsin*x)^2 - Int(x*`.`*``(2*arcsin*x/sqrt(1-x ^2)),x)" "6#/%!G,&*&%\"xG\"\"\"*$*&%'arcsinGF(F'F(\"\"#F(F(-%$IntG6$*( F'F(%\".GF(-F$6#**F,F(F+F(F'F(-%%sqrtG6#,&F(F(*$F'F,!\"\"F:F(F'F:" } {TEXT -1 1 " " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "`` = x*(arcsin*x)^2-Int(arcsin*x*`.`*``(2*x/sqrt(1-x^2)),x)" "6#/%!G,&*&% \"xG\"\"\"*$*&%'arcsinGF(F'F(\"\"#F(F(-%$IntG6$**F+F(F'F(%\".GF(-F$6#* (F,F(F'F(-%%sqrtG6#,&F(F(*$F'F,!\"\"F:F(F'F:" }{TEXT -1 1 " " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "``=x*(arcsin*x)^2+2*Int(arc sin*x*`.`*``(-x/sqrt(1-x^2)),x)" "6#/%!G,&*&%\"xG\"\"\"*$*&%'arcsinGF( F'F(\"\"#F(F(*&F,F(-%$IntG6$**F+F(F'F(%\".GF(-F$6#,$*&F'F(-%%sqrtG6#,& F(F(*$F'F,!\"\"F " 0 "" {MPLTEXT 1 0 155 "f := x -> x*arcsin(x)^2+2*arcsin(x)*sqrt(1-x^2)-2 *x:\n'f(x)'=f(x);\ng := x -> arcsin(x)^2:\n'g(x)'=g(x);\nplot([f(x),g( x)],x=-1..1,-.47..1.2,color=[red,blue]);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%\"fG6#%\"xG,(*&F'\"\"\")-%'arcsinGF&\"\"#F*F**(F.F*F,F*,&F*F* *$)F'F.F*!\"\"#F*F.F**&F.F*F'F*F3" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/ -%\"gG6#%\"xG*$)-%'arcsinGF&\"\"#\"\"\"" }}{PARA 13 "" 1 "" {GLPLOT2D 404 314 314 {PLOTDATA 2 "6&-%'CURVESG6$7S7$$!\"\"\"\"!$!3)\\RBF+6Sn%!# =7$$!3ommm;p0k&*F-$!3sx$pt(yX\\QF-7$$!3wKL$3zv#F-7$$!3:mmm\"4m(G$)F-$!3!*ekI79u1BF-7$$!3 \"QLL3i.9!zF-$!3n0cK#>F-7$$!3\"ommT!R=0vF-$!3mO]\")>#3mh\"F-7$$!3u ****\\P8#\\4(F-$!3Y]?Gh#R;M\"F-7$$!3+nm;/siqmF-$!3e`3cw6$p4\"F-7$$!3[+ +](y$pZiF-$!3KQ1l\"HJ?)))!#>7$$!33LLL$yaE\"eF-$!3uCwpZbudqFen7$$!3hmmm \">s%HaF-$!3[UH[2IS\"p&Fen7$$!3Q+++]$*4)*\\F-$!3S@S#faVDR%Fen7$$!39+++ ]_&\\c%F-$!39$*)z3KP[J$Fen7$$!31+++]1aZTF-$!3;El!4:3fY#Fen7$$!3umm;/#) [oPF-$!3@rU[(ziv$=Fen7$$!3hLLL$=exJ$F-$!3gU`s!pK`C\"Fen7$$!3*RLLLtIf$H F-$!3yj'\\\"R,-'e)!#?7$$!3]++]PYx\"\\#F-$!3l>*\\w\\(pA_F^q7$$!3EMLLL7i )4#F-$!3T9hY=t_3JF^q7$$!3c****\\P'psm\"F-$!3![_e\"[lc`:F^q7$$!3')**** \\74_c7F-$!3v0Cyii%Qj'!#@7$$!3)3LLL3x%z#)Fen$!3[?z3xQX%*=Fcr7$$!3KMLL3 s$QM%Fen$!3o[*3]D\\Jt#!#A7$$!3]^omm;zr)*Fcr$!36)=4$QBw1K!#F7$$\"3%pJL$ ezw5VFen$\"3?rR>!G'=rEF^s7$$\"3s*)***\\PQ#\\\")Fen$\"3E)e`V>uj!=Fcr7$$ \"3GKLLe\"*[H7F-$\"3oK,@GJ+9iFcr7$$\"3I*******pvxl\"F-$\"3xs-L4+3F:F^q 7$$\"3#z****\\_qn2#F-$\"3mkH%>#R(=,$F^q7$$\"3U)***\\i&p@[#F-$\"3#e;9x& G-i^F^q7$$\"3B)****\\2'HKHF-$\"33HY =')=n7Fen7$$\"3i******\\2goPF-$\"30C&pA$3tP=Fen7$$\"3UKL$eR<*fTF-$\"3) oFvWx<')[#Fen7$$\"3m******\\)Hxe%F-$\"3yMfWmYHmLFen7$$\"3ckm;H!o-*\\F- $\"3MZ%)fg&G6P%Fen7$$\"3y)***\\7k.6aF-$\"3QPgK\\b!4j&Fen7$$\"3#emmmT9C #eF-$\"3Ab-D7&o`4(Fen7$$\"33****\\i!*3`iF-$\"3obb3^ni1*)Fen7$$\"3%QLLL $*zym'F-$\"3;-a#f&oY&4\"F-7$$\"3wKLL3N1#4(F-$\"3(Q7O27i)R8F-7$$\"3Nmm; HYt7vF-$\"3'zXKu9c?i\"F-7$$\"3Y*******p(G**yF-$\"3*Gu\\7n+:#>F-7$$\"3] mmmT6KU$)F-$\"3))4aBF-7$$\"3fKLLLbdQ()F-$\"3h0vk:5pNFF-7$$\"3[++ ]i`1h\"*F-$\"3_*zHs_edD$F-7$$\"3W++]P?Wl&*F-$\"3SIEy4%3<&QF-7$$\"\"\"F *$\"3)\\RBF+6Sn%F--%'COLOURG6&%$RGBG$\"*++++\"!\")$F*F*Fd[l-F$6$7_o7$F ($\"3]RBF+6SnC!#<7$$!3-n;HdNvs**F-$\"3lQ#3r,!*3C#F[\\l7$$!3/MLe9r]X**F -$\"3;DY=!)**=]@F[\\l7$$!3/,](=ng#=**F-$\"3A$G+H*p\"=3#F[\\l7$$!3%pmm \"HU,\"*)*F-$\"3b35qe`*\\-#F[\\l7$$!3()***\\PM@l$)*F-$\"3zPnT0![8$>F[ \\l7$$!3!RLL$e%G?y*F-$\"3unREpj,a=F[\\l7$$!3u****\\(oUIn*F-$\"36^'R'[l eF$\"37n1lP1<)o*F-7$FC$\"3kVUCswbmF^q7$F`t$\"35mVh*[A$>:Fen7$Fet$\"3g\"*GGT;xtF Fen7$Fjt$\"3;ufQt([kP%Fen7$F_u$\"3!=,HC-H?H'Fen7$Fdu$\"3%z)3%)Qjwc))Fe n7$Fiu$\"3mK&yqCft:\"F-7$F^v$\"3Y]0h673$\\\"F-7$Fcv$\"3ejZ[6TpS=F-7$Fh v$\"3CAT`sqgrAF-7$F]w$\"3B;cg:`!)HFF-7$Fbw$\"3OMiB!Ho*oKF-7$Fgw$\"3#4[ =lD5C'QF-7$F\\x$\"3C(fkt\\rLc%F-7$Fax$\"3P'*3K:1SF`F-7$Ffx$\"3Z#[[uS,` @'F-7$F[y$\"3t)ffF=@[A(F-7$F`y$\"3yG$*\\dUh$H)F-7$Fey$\"3kGg')fL`O(*F- 7$Fjy$\"3;-9(y?T,8\"F[\\l7$$\"3amm\"zW?)\\*)F-$\"38\"*zf-0`G7F[\\l7$F_ z$\"3jLsX'Hk:M\"F[\\l7$$\"3Y++++PDj$*F-$\"3e3?D1*z*o9F[\\l7$Fdz$\"3m'* 34DmRD;F[\\l7$$\"3K+]7G:3u'*F-$\"3,OEo:VmGF[\\l7$$\"35+]P40O\"*)* F-$\"3(zP<%)Hlc-#F[\\l7$$\"3k]7.#Q?&=**F-$\"3Y$4kC&[S#3#F[\\l7$$\"31+v oa-oX**F-$\"3WL-NWun]@F[\\l7$$\"3[\\PMF,%G(**F-$\"3CSJGb;CTAF[\\l7$Fiz Fi[l-F^[l6&F`[lFd[lFd[lFa[l-%+AXESLABELSG6$Q\"x6\"Q!F[[m-%%VIEWG6$;F(F iz;$!#Z!\"#$\"#7F)" 1 2 0 1 10 0 2 9 1 4 2 1.000000 45.000000 45.000000 0 0 "Curve 1" "Curve 2" }}}}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 116 "Int(arcsin(x)^2,x);\n``=stu dent[intparts](%,arcsin(x)^2);\n``=student[intparts](rhs(%),arcsin(x)) ;\n``=value(rhs(%))+c;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%$IntG6$*$) -%'arcsinG6#%\"xG\"\"#\"\"\"F+" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/%!G ,&*&)-%'arcsinG6#%\"xG\"\"#\"\"\"F+F-F--%$IntG6$,$**F,F-F(F-,&F-F-*$)F +F,F-!\"\"#F6F,F+F-F-F+F6" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/%!G,(*&) -%'arcsinG6#%\"xG\"\"#\"\"\"F+F-F-*(F,F-F(F-,&F-F-*$)F+F,F-!\"\"#F-F,F --%$IntG6$!\"#F+F-" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/%!G,**&)-%'arcs inG6#%\"xG\"\"#\"\"\"F+F-F-*(F,F-F(F-,&F-F-*$)F+F,F-!\"\"#F-F,F-*&F,F- F+F-F2%\"cGF-" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 106 "More details of the first aplication of the integration \+ by parts formula are shown by using the procedure " }{TEXT 0 10 "intby parts" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 63 "Int(arcsin(x)^2,x);\n``=intbyparts( %,u=arcsin(x)^2,v,info=true);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%$In tG6$*$)-%'arcsinG6#%\"xG\"\"#\"\"\"F+" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%*PIECEWISEG6$7$/%\"uG*$)-%'arcsinG6#%\"xG\"\"#\"\"\"/%\"vGF.7$/** %\"dGF0F(F0%#~dG!\"\"F.F8,$*(F/F0F+F0,&F0F0*$)F.F/F0F8#F8F/F0/**F6F0F2 F0F7F8F.F8F0" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%$IntG6$*$)-%'arcsin G6#%\"xG\"\"#\"\"\"F,-F%6$*&%\"uGF.-%!G6#**%\"dGF.%\"vGF.%#~dG!\"\"F,F :F.F," }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/%!G,&*&%\"uG\"\"\"%\"vGF(F(- %$IntG6$*&F)F(-F$6#**%\"dGF(F'F(%#~dG!\"\"%\"xGF3F(F4F3" }}{PARA 11 " " 1 "" {XPPMATH 20 "6#/%!G,&*&)-%'arcsinG6#%\"xG\"\"#\"\"\"F+F-F--%$In tG6$,$**F,F-F+F-F(F-,&F-F-*$)F+F,F-!\"\"#F6F,F-F+F6" }}}{PARA 0 "" 0 " " {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 107 "More details of the se cond aplication of the integration by parts formula are shown by using the procedure " }{TEXT 0 10 "intbyparts" }{TEXT -1 2 ". " }}{PARA 0 " " 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 116 "Int( 2*x*arcsin(x)/sqrt(1-x^2),x);\n``=intbyparts(%,u=arcsin(x),v,info=true );\n``=simplify(rhs(%));\n``=value(rhs(%))+c;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%$IntG6$,$**\"\"#\"\"\"%\"xGF)-%'arcsinG6#F*F),&F)F)*$ )F*F(F)!\"\"#F1F(F)F*" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%*PIECEWISEG 6$7$/%\"uG-%'arcsinG6#%\"xG/%\"vG,$*&\"\"#\"\"\",&F2F2*$)F,F1F2!\"\"#F 2F1F67$/**%\"dGF2F(F2%#~dGF6F,F6*&F2F2*$F3#F2F1F6/**F;F2F.F2F " 0 "" {MPLTEXT 1 0 1 ";" }}}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 11 "Example 10 " }}{PARA 258 "" 0 "" {TEXT 293 8 "Question" }{TEXT -1 13 ": Find (a) " }{XPPEDIT 18 0 "Int(sqrt (1-x^2),x);" "6#-%$IntG6$-%%sqrtG6#,&\"\"\"F**$%\"xG\"\"#!\"\"F," } {TEXT -1 10 " and (b) " }{XPPEDIT 18 0 "Int(x^2/sqrt(1-x^2),x)" "6#-% $IntG6$*&%\"xG\"\"#-%%sqrtG6#,&\"\"\"F-*$F'F(!\"\"F/F'" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT 294 8 "Solution" }{TEXT -1 2 ": " }}{PARA 0 "" 0 "" {TEXT -1 68 "(a) This integral can be found by making the (i nverse) substitution " }{XPPEDIT 18 0 "x=sin*theta" "6#/%\"xG*&%$sinG \"\"\"%&thetaGF'" }{TEXT -1 72 ", but an alternative method is to use \+ the integration by parts formula. " }}{PARA 256 "" 0 "" {TEXT -1 1 " \+ " }{XPPEDIT 18 0 "Int(sqrt(1-x^2),x)" "6#-%$IntG6$-%%sqrtG6#,&\"\"\"F* *$%\"xG\"\"#!\"\"F," }{TEXT -1 8 " ... " }{XPPEDIT 18 0 "PIECEWISE( [u=sqrt(1-x^2),v=x],[du/dx=-x/sqrt(1-x^2),dv/dx=1])" "6#-%*PIECEWISEG6 $7$/%\"uG-%%sqrtG6#,&\"\"\"F-*$%\"xG\"\"#!\"\"/%\"vGF/7$/*&%#duGF-%#dx GF1,$*&F/F--F*6#,&F-F-*$F/F0F1F1F1/*&%#dvGF-F8F1F-" }{TEXT -1 1 " " }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 256 "" 0 "" {TEXT -1 1 " " } {XPPEDIT 18 0 "``= Int(u*``(dv/dx),x)" "6#/%!G-%$IntG6$*&%\"uG\"\"\"-F $6#*&%#dvGF*%#dxG!\"\"F*%\"xG" }{TEXT -1 1 " " }}{PARA 256 "" 0 "" {TEXT -1 2 " " }{XPPEDIT 18 0 "`` = u*v-Int(v*``(du/dx),x)" "6#/%!G,& *&%\"uG\"\"\"%\"vGF(F(-%$IntG6$*&F)F(-F$6#*&%#duGF(%#dxG!\"\"F(%\"xGF3 " }{TEXT -1 1 " " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "` `=x*sqrt(1-x^2)-Int(-x^2/sqrt(1-x^2),x)" "6#/%!G,&*&%\"xG\"\"\"-%%sqrt G6#,&F(F(*$F'\"\"#!\"\"F(F(-%$IntG6$,$*&F'F.-F*6#,&F(F(*$F'F.F/F/F/F'F /" }{TEXT -1 1 " " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 " `` = x*sqrt(1-x^2)-Int((1-x^2)/sqrt(1-x^2),x)+Int(1/sqrt(1-x^2),x);" " 6#/%!G,(*&%\"xG\"\"\"-%%sqrtG6#,&F(F(*$F'\"\"#!\"\"F(F(-%$IntG6$*&,&F( F(*$F'F.F/F(-F*6#,&F(F(*$F'F.F/F/F'F/-F16$*&F(F(-F*6#,&F(F(*$F'F.F/F/F 'F(" }{TEXT -1 1 " " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "`` = x*sqrt(1-x^2)-Int(sqrt(1-x^2),x)+Int(1/sqrt(1-x^2),x);" "6#/%! G,(*&%\"xG\"\"\"-%%sqrtG6#,&F(F(*$F'\"\"#!\"\"F(F(-%$IntG6$-F*6#,&F(F( *$F'F.F/F'F/-F16$*&F(F(-F*6#,&F(F(*$F'F.F/F/F'F(" }{TEXT -1 2 ". " }} {PARA 0 "" 0 "" {TEXT -1 6 "Hence " }}{PARA 256 "" 0 "" {TEXT -1 1 " \+ " }{XPPEDIT 18 0 "2*Int(sqrt(1-x^2),x) = x*sqrt(1-x^2)+Int(1/sqrt(1-x^ 2),x);" "6#/*&\"\"#\"\"\"-%$IntG6$-%%sqrtG6#,&F&F&*$%\"xGF%!\"\"F/F&,& *&F/F&-F+6#,&F&F&*$F/F%F0F&F&-F(6$*&F&F&-F+6#,&F&F&*$F/F%F0F0F/F&" } {TEXT -1 1 " " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "`` = x*sqrt(1-x^2)+arcsin*x+c;" "6#/%!G,(*&%\"xG\"\"\"-%%sqrtG6#,&F(F(*$F' \"\"#!\"\"F(F(*&%'arcsinGF(F'F(F(%\"cGF(" }{TEXT -1 2 ", " }}{PARA 0 " " 0 "" {TEXT -1 8 "so that " }}{PARA 256 "" 0 "" {TEXT -1 1 " " } {XPPEDIT 18 0 "Int(sqrt(1-x^2),x)=1/2" "6#/-%$IntG6$-%%sqrtG6#,&\"\"\" F+*$%\"xG\"\"#!\"\"F-*&F+F+F.F/" }{TEXT -1 1 " " }{XPPEDIT 18 0 "x*sqr t(1-x^2)+1/2;" "6#,&*&%\"xG\"\"\"-%%sqrtG6#,&F&F&*$F%\"\"#!\"\"F&F&*&F &F&F,F-F&" }{TEXT -1 1 " " }{XPPEDIT 18 0 "arcsin*x+c[1]" "6#,&*&%'arc sinG\"\"\"%\"xGF&F&&%\"cG6#F&F&" }{TEXT -1 2 ", " }}{PARA 0 "" 0 "" {TEXT -1 6 "where " }{XPPEDIT 18 0 "c[1]=c/2" "6#/&%\"cG6#\"\"\"*&F%F' \"\"#!\"\"" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 258 "" 0 "" {TEXT -1 5 "Let " } {XPPEDIT 18 0 "f(x) = 1/2;" "6#/-%\"fG6#%\"xG*&\"\"\"F)\"\"#!\"\"" } {TEXT -1 1 " " }{XPPEDIT 18 0 "x*sqrt(1-x^2)+1/2;" "6#,&*&%\"xG\"\"\"- %%sqrtG6#,&F&F&*$F%\"\"#!\"\"F&F&*&F&F&F,F-F&" }{TEXT -1 1 " " } {XPPEDIT 18 0 "arcsin*x;" "6#*&%'arcsinG\"\"\"%\"xGF%" }{TEXT -1 6 " a nd " }{XPPEDIT 18 0 "g(x) = sqrt(1-x^2);" "6#/-%\"gG6#%\"xG-%%sqrtG6# ,&\"\"\"F,*$F'\"\"#!\"\"" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 39 "In the following picture the graph of " }{XPPEDIT 18 0 "f(x)" "6#-%\"fG6#%\"xG" }{TEXT -1 13 " is drawn in " }{TEXT 260 3 "red" }{TEXT -1 20 " while the graph of " } {XPPEDIT 18 0 "g(x);" "6#-%\"gG6#%\"xG" }{TEXT -1 13 " is drawn in " } {TEXT 256 4 "blue" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 45 "The picture is consistent with the fact that " }{XPPEDIT 18 0 "Diff([f(x) ],x) = g(x);" "6#/-%%DiffG6$7#-%\"fG6#%\"xGF+-%\"gG6#F+" }{TEXT -1 2 " . " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 131 "f := x -> x*sqrt(1-x^2)/2+a rcsin(x)/2:\n'f(x)'=f(x);\ng := x -> sqrt(1-x^2):\n'g(x)'=g(x);\nplot( [f(x),g(x)],x=-1..1,color=[red,blue]);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%\"fG6#%\"xG,&*(\"\"#!\"\"F'\"\"\",&F,F,*$)F'F*F,F+#F,F*F,*&F0 F,-%'arcsinGF&F,F," }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%\"gG6#%\"xG*$ ,&\"\"\"F**$)F'\"\"#F*!\"\"#F*F-" }}{PARA 13 "" 1 "" {GLPLOT2D 372 330 330 {PLOTDATA 2 "6&-%'CURVESG6$7S7$$!\"\"\"\"!$!3!G[uRj\")R&y!#=7$ $!3ommm;p0k&*F-$!3%oT2#p*G(oxF-7$$!3wKL$3s%H aF-$!3%>Ww\"o)e%\\^F-7$$!3Q+++]$*4)*\\F-$!3GmM+s7T\"y%F-7$$!39+++]_&\\ c%F-$!3Nv$33*3/,WF-7$$!31+++]1aZTF-$!3myAfNdNDSF-7$$!3umm;/#)[oPF-$!3? 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" }{XPPEDIT 18 0 "PIECEWISE([u = -x, v \+ = sqrt(1-x^2)],[du/dx = -1, dv/dx = -x/sqrt(1-x^2)]);" "6#-%*PIECEWISE G6$7$/%\"uG,$%\"xG!\"\"/%\"vG-%%sqrtG6#,&\"\"\"F2*$F*\"\"#F+7$/*&%#duG F2%#dxGF+,$F2F+/*&%#dvGF2F9F+,$*&F*F2-F/6#,&F2F2*$F*F4F+F+F+" }{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "``= Int(u*``(dv/dx),x)" "6#/%!G-%$IntG6$*&%\"uG \"\"\"-F$6#*&%#dvGF*%#dxG!\"\"F*%\"xG" }{TEXT -1 1 " " }}{PARA 256 "" 0 "" {TEXT -1 2 " " }{XPPEDIT 18 0 "`` = u*v-Int(v*``(du/dx),x)" "6#/ %!G,&*&%\"uG\"\"\"%\"vGF(F(-%$IntG6$*&F)F(-F$6#*&%#duGF(%#dxG!\"\"F(% \"xGF3" }{TEXT -1 1 " " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "`` = -x*sqrt(1-x^2)+Int(sqrt(1-x^2),x);" "6#/%!G,&*&%\"xG\"\"\"- %%sqrtG6#,&F(F(*$F'\"\"#!\"\"F(F/-%$IntG6$-F*6#,&F(F(*$F'F.F/F'F(" } {TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 13 "The equation " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "Int(x^2/sqrt(1-x^2),x)=-x*s qrt(1-x^2)+Int(sqrt(1-x^2),x)" "6#/-%$IntG6$*&%\"xG\"\"#-%%sqrtG6#,&\" \"\"F.*$F(F)!\"\"F0F(,&*&F(F.-F+6#,&F.F.*$F(F)F0F.F0-F%6$-F+6#,&F.F.*$ F(F)F0F(F." }{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 58 "can also be obtained from the solution given for \+ part (a)." }}{PARA 0 "" 0 "" {TEXT -1 47 "Making use of the result fro m part (a) we have " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "Int(x^2/sqrt(1-x^2),x)=-1/2" "6#/-%$IntG6$*&%\"xG\"\"#-%%sqrtG6#,& \"\"\"F.*$F(F)!\"\"F0F(,$*&F.F.F)F0F0" }{TEXT -1 1 " " }{XPPEDIT 18 0 "x*sqrt(1-x^2)+1/2;" "6#,&*&%\"xG\"\"\"-%%sqrtG6#,&F&F&*$F%\"\"#!\"\"F &F&*&F&F&F,F-F&" }{TEXT -1 1 " " }{XPPEDIT 18 0 "arcsin*x+c;" "6#,&*&% 'arcsinG\"\"\"%\"xGF&F&%\"cGF&" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 258 "" 0 "" {TEXT -1 5 "Let " }{XPPEDIT 18 0 "f (x) = -1/2;" "6#/-%\"fG6#%\"xG,$*&\"\"\"F*\"\"#!\"\"F," }{TEXT -1 1 " \+ " }{XPPEDIT 18 0 "x*sqrt(1-x^2)+1/2;" "6#,&*&%\"xG\"\"\"-%%sqrtG6#,&F& F&*$F%\"\"#!\"\"F&F&*&F&F&F,F-F&" }{TEXT -1 1 " " }{XPPEDIT 18 0 "arcs in*x;" "6#*&%'arcsinG\"\"\"%\"xGF%" }{TEXT -1 6 " and " }{XPPEDIT 18 0 "g(x) = x^2/sqrt(1-x^2);" "6#/-%\"gG6#%\"xG*&F'\"\"#-%%sqrtG6#,&\"\" \"F.*$F'F)!\"\"F0" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {PARA 0 "" 0 "" {TEXT -1 39 "In the following picture the graph of " }{XPPEDIT 18 0 "f(x)" "6#-%\"fG6#%\"xG" }{TEXT -1 13 " is drawn in " } {TEXT 260 3 "red" }{TEXT -1 20 " while the graph of " }{XPPEDIT 18 0 " g(x);" "6#-%\"gG6#%\"xG" }{TEXT -1 13 " is drawn in " }{TEXT 256 4 "bl ue" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 45 "The picture is con sistent with the fact that " }{XPPEDIT 18 0 "Diff([f(x)],x) = g(x);" " 6#/-%%DiffG6$7#-%\"fG6#%\"xGF+-%\"gG6#F+" }{TEXT -1 2 ". " }}{PARA 0 " " 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 156 "f := x -> -x*sqrt(1-x^2)/2+arcsin(x)/2:\n' f(x)'=f(x);\ng := x -> x^2/sqrt(1-x^2):\n'g(x)'=g(x);\nplot([f(x),g(x) ],x=-1..1,-.8..3,color=[red,blue],xtickmarks=4);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%\"fG6#%\"xG,&*(\"\"#!\"\"F'\"\"\",&F,F,*$)F'F*F,F+#F ,F*F+*&F0F,-%'arcsinGF&F,F," }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%\"gG 6#%\"xG*&,&\"\"\"F**$)F'\"\"#F*!\"\"#F.F-F'F-" }}{PARA 13 "" 1 "" {GLPLOT2D 271 354 354 {PLOTDATA 2 "6'-%'CURVESG6$7en7$$!\"\"\"\"!$!3!G [uRj\")R&y!#=7$$!3%pmm\"HU,\"*)*F-$!3q*Qj$z4'pQ'F-7$$!3!RLL$e%G?y*F-$! 39**)e8\\'[#z&F-7$$!3u****\\(oUIn*F-$!3m>(\\f,__M&F-7$$!3ommm;p0k&*F-$ !31/+KH)Gc(\\F-7$$!3E++vV5Su$*F-$!3DGj]BB=WWF-7$$!3wKL$3\\n)G4AKF-7$$!3:mmm\"4m(G$)F-$!3Kx.Z` 6h;EF-7$$!3\"QLL3i.9!zF-$!3.K@.8AqL@F-7$$!3\"ommT!R=0vF-$!3#HhZv/KVw\" F-7$$!3u****\\P8#\\4(F-$!3]U7$$!33LLL$yaE\"eF-$!3]G:%H\\e\\O(Fio7 $$!3hmmm\">s%HaF-$!3%[\"*pga#o**eFio7$$!3Q+++]$*4)*\\F-$!3^Y259%>Q_%Fi o7$$!39+++]_&\\c%F-$!3Q9$\\b[G[R$Fio7$$!31+++]1aZTF-$!35`wDyMp8DFio7$$ !3umm;/#)[oPF-$!3KB'H%4\">j'=Fio7$$!3hLLL$=exJ$F-$!3]L%4]!p4g7Fio7$$!3 *RLLLtIf$HF-$!39.Vc1KTk')!#?7$$!3]++]PYx\"\\#F-$!3UO*=A!Q[c_Fbr7$$!3EM LLL7i)4#F-$!3$3]z%)\\>E7$Fbr7$$!3c****\\P'psm\"F-$!3O)=ne9kzb\"Fbr7$$! 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" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 98 "Int(x^2/sqrt(1-x^2),x);\n``=intbyparts(%,u=x,v,i nfo=true);\n``=simplify(rhs(%));\n``=value(rhs(%))+c;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%$IntG6$*&%\"xG\"\"#,&\"\"\"F**$)F'F(F*!\"\"#F-F( F'" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%*PIECEWISEG6$7$/%\"uG%\"xG/%\" vG,$*$,&\"\"\"F/*$)F)\"\"#F/!\"\"#F/F2F37$/**%\"dGF/F(F/%#~dGF3F)F3F// **F8F/F+F/F9F3F)F3*&F)F/F.#F3F2" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-% $IntG6$*&%\"xG\"\"#,&\"\"\"F+*$)F(F)F+!\"\"#F.F)F(-F%6$*&%\"uGF+-%!G6# **%\"dGF+%\"vGF+%#~dGF.F(F.F+F(" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/%! G,&*&%\"uG\"\"\"%\"vGF(F(-%$IntG6$*&F)F(-F$6#**%\"dGF(F'F(%#~dG!\"\"% \"xGF3F(F4F3" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/%!G,&*&%\"xG\"\"\",&F (F(*$)F'\"\"#F(!\"\"#F(F,F--%$IntG6$,$*$F)F.F-F'F-" }}{PARA 11 "" 1 " " {XPPMATH 20 "6#/%!G,&*&%\"xG\"\"\",&F(F(*$)F'\"\"#F(!\"\"#F(F,F--%$I ntG6$*$F)F.F'F(" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/%!G,(*(\"\"#!\"\"% \"xG\"\"\",&F*F**$)F)F'F*F(#F*F'F(*&F.F*-%'arcsinG6#F)F*F*%\"cGF*" }}} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 5 "Tasks" }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{SECT 1 {PARA 4 "" 0 " " {TEXT -1 3 "Q1 " }}{PARA 0 "" 0 "" {TEXT -1 6 "Find " }{XPPEDIT 18 0 "Int(x*exp(-x),x);" "6#-%$IntG6$*&%\"xG\"\"\"-%$expG6#,$F'!\"\"F(F' " }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{SECT 1 {PARA 5 " " 0 "" {TEXT -1 4 "Ans " }}{PARA 0 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "-x*exp(-x)-exp(-x)+c" "6#,(*&%\"xG\"\"\"-%$expG6#,$F%!\"\"F&F+-F (6#,$F%F+F+%\"cGF&" }{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 85 "Int(x*exp(-x),x);\n``=studen t[intparts](%,x);\n``=simplify(rhs(%));\n``=value(rhs(%))+c;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%$IntG6$*&%\"xG\"\"\"-%$expG6#,$F'!\"\"F(F' " }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/%!G,&*&%\"xG\"\"\"-%$expG6#,$F'! \"\"F(F--%$IntG6$,$F)F-F'F-" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/%!G,&* &%\"xG\"\"\"-%$expG6#,$F'!\"\"F(F--%$IntG6$F)F'F(" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/%!G,(*&%\"xG\"\"\"-%$expG6#,$F'!\"\"F(F-F)F-%\"cGF(" } }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 46 "More de tails are shown by using the procedure " }{TEXT 0 10 "intbyparts" } {TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 92 "Int(x*exp(-x),x);\n``=intbyparts(%,u=x,v,info=tr ue);\n``=simplify(rhs(%));\n``=value(rhs(%))+c;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%$IntG6$*&%\"xG\"\"\"-%$expG6#,$F'!\"\"F(F'" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%*PIECEWISEG6$7$/%\"uG%\"xG/%\"vG,$-%$expG6 #,$F)!\"\"F17$/**%\"dG\"\"\"F(F6%#~dGF1F)F1F6/**F5F6F+F6F7F1F)F1F-" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#/-%$IntG6$*&%\"xG\"\"\"-%$expG6#,$F(! \"\"F)F(-F%6$*&%\"uGF)-%!G6#**%\"dGF)%\"vGF)%#~dGF.F(F.F)F(" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/%!G,&*&%\"uG\"\"\"%\"vGF(F(-%$IntG6$*&F)F(- F$6#**%\"dGF(F'F(%#~dG!\"\"%\"xGF3F(F4F3" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/%!G,&*&%\"xG\"\"\"-%$expG6#,$F'!\"\"F(F--%$IntG6$,$F)F-F'F-" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#/%!G,&*&%\"xG\"\"\"-%$expG6#,$F'!\"\"F (F--%$IntG6$F)F'F(" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/%!G,(*&%\"xG\" \"\"-%$expG6#,$F'!\"\"F(F-F)F-%\"cGF(" }}}{PARA 0 "" 0 "" {TEXT -1 0 " " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{PARA 0 "" 0 "" {TEXT -1 27 "___________________________" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 27 "___________________________" }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{SECT 1 {PARA 4 "" 0 " " {TEXT -1 3 "Q2 " }}{PARA 0 "" 0 "" {TEXT -1 6 "Find " }{XPPEDIT 18 0 "Int(x^2*exp(-x),x)" "6#-%$IntG6$*&%\"xG\"\"#-%$expG6#,$F'!\"\"\"\" \"F'" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{SECT 1 {PARA 5 "" 0 "" {TEXT -1 4 "Ans " }}{PARA 0 "" 0 "" {TEXT -1 1 " " } {XPPEDIT 18 0 "-x^2*exp(-x)-2*x*exp(-x)-2*exp(-x)+c" "6#,**&%\"xG\"\"# -%$expG6#,$F%!\"\"\"\"\"F+*(F&F,F%F,-F(6#,$F%F+F,F+*&F&F,-F(6#,$F%F+F, F+%\"cGF," }{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 100 "Int(x^2*exp(-x),x);\n``=student[in tparts](%,x^2);\n``=student[intparts](rhs(%),x);\n``=value(rhs(%))+c; " }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%$IntG6$*&)%\"xG\"\"#\"\"\"-%$exp G6#,$F(!\"\"F*F(" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/%!G,&*&)%\"xG\"\" #\"\"\"-%$expG6#,$F(!\"\"F*F/-%$IntG6$,$*(F)F*F(F*F+F*F/F(F/" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/%!G,(*&)%\"xG\"\"#\"\"\"-%$expG6#,$F(!\"\"F *F/*(F)F*F(F*F+F*F/-%$IntG6$,$*&F)F*F+F*F*F(F*" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/%!G,**&)%\"xG\"\"#\"\"\"-%$expG6#,$F(!\"\"F*F/*(F)F*F( F*F+F*F/*&F)F*F+F*F/%\"cGF*" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 107 "More details of the first application of the i ntegration by parts formula are shown by using the procedure " }{TEXT 0 10 "intbyparts" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 76 "Int(x^2*exp(-x),x);\n``=intb yparts(%,u=x^2,v,info=true);\n``=simplify(rhs(%));" }}{PARA 11 "" 1 " " {XPPMATH 20 "6#-%$IntG6$*&)%\"xG\"\"#\"\"\"-%$expG6#,$F(!\"\"F*F(" } }{PARA 11 "" 1 "" {XPPMATH 20 "6#-%*PIECEWISEG6$7$/%\"uG*$)%\"xG\"\"# \"\"\"/%\"vG,$-%$expG6#,$F+!\"\"F57$/**%\"dGF-F(F-%#~dGF5F+F5,$*&F,F-F +F-F-/**F9F-F/F-F:F5F+F5F1" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%$IntG 6$*&)%\"xG\"\"#\"\"\"-%$expG6#,$F)!\"\"F+F)-F%6$*&%\"uGF+-%!G6#**%\"dG F+%\"vGF+%#~dGF0F)F0F+F)" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/%!G,&*&% \"uG\"\"\"%\"vGF(F(-%$IntG6$*&F)F(-F$6#**%\"dGF(F'F(%#~dG!\"\"%\"xGF3F (F4F3" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/%!G,&*&)%\"xG\"\"#\"\"\"-%$e xpG6#,$F(!\"\"F*F/-%$IntG6$,$*(F)F*F(F*F+F*F/F(F/" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/%!G,&*&)%\"xG\"\"#\"\"\"-%$expG6#,$F(!\"\"F*F/*&F)F*-% $IntG6$*&F(F*F+F*F(F*F*" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{PARA 0 "" 0 "" {TEXT -1 27 " ___________________________" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" } }{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 " " {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 27 "___________________________" }}{EXCHG {PARA 0 " > " 0 "" {MPLTEXT 1 0 1 ";" }}}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 3 "Q3 " }}{PARA 0 "" 0 "" {TEXT -1 6 "Find " }{XPPEDIT 18 0 "Int(x*cos*3*x ,x);" "6#-%$IntG6$**%\"xG\"\"\"%$cosGF(\"\"$F(F'F(F'" }{TEXT -1 2 ". \+ " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{SECT 1 {PARA 5 "" 0 "" {TEXT -1 4 "Ans " }}{PARA 0 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "x/3" "6#*&% \"xG\"\"\"\"\"$!\"\"" }{TEXT -1 1 " " }{XPPEDIT 18 0 "sin*3*x+1/9;" "6 #,&*(%$sinG\"\"\"\"\"$F&%\"xGF&F&*&F&F&\"\"*!\"\"F&" }{TEXT -1 1 " " } {XPPEDIT 18 0 "cos*3*x+c;" "6#,&*(%$cosG\"\"\"\"\"$F&%\"xGF&F&%\"cGF& " }{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> \+ " 0 "" {MPLTEXT 1 0 65 "Int(x*cos(3*x),x);\n``=student[intparts](%,x); \n``=value(rhs(%))+c;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%$IntG6$*&% \"xG\"\"\"-%$cosG6#,$*&\"\"$F(F'F(F(F(F'" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/%!G,&*&#\"\"\"\"\"$F(*&%\"xGF(-%$sinG6#,$*&F)F(F+F(F(F(F(F(-%$I ntG6$,$*&F'F(F,F(F(F+!\"\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/%!G,(*& #\"\"\"\"\"$F(*&%\"xGF(-%$sinG6#,$*&F)F(F+F(F(F(F(F(*&#F(\"\"*F(-%$cos GF.F(F(%\"cGF(" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 46 "More details are shown by using the procedure " }{TEXT 0 10 "intbyparts" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 93 "Int(x*cos(3*x),x);\n``=intby parts(%,u=x,v,info=true);\n``=simplify(rhs(%));\n``=value(rhs(%))+c;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%$IntG6$*&%\"xG\"\"\"-%$cosG6#,$*& \"\"$F(F'F(F(F(F'" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%*PIECEWISEG6$7$ /%\"uG%\"xG/%\"vG,$*&#\"\"\"\"\"$F/-%$sinG6#,$*&F0F/F)F/F/F/F/7$/**%\" dGF/F(F/%#~dG!\"\"F)F;F//**F9F/F+F/F:F;F)F;-%$cosGF3" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%$IntG6$*&%\"xG\"\"\"-%$cosG6#,$*&\"\"$F)F(F)F)F)F (-F%6$*&%\"uGF)-%!G6#**%\"dGF)%\"vGF)%#~dG!\"\"F(F;F)F(" }}{PARA 11 " " 1 "" {XPPMATH 20 "6#/%!G,&*&%\"uG\"\"\"%\"vGF(F(-%$IntG6$*&F)F(-F$6# **%\"dGF(F'F(%#~dG!\"\"%\"xGF3F(F4F3" }}{PARA 11 "" 1 "" {XPPMATH 20 " 6#/%!G,&*&#\"\"\"\"\"$F(*&%\"xGF(-%$sinG6#,$*&F)F(F+F(F(F(F(F(-%$IntG6 $,$*&F'F(F,F(F(F+!\"\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/%!G,&*&#\" \"\"\"\"$F(*&%\"xGF(-%$sinG6#,$*&F)F(F+F(F(F(F(F(*&#F(F)F(-%$IntG6$F,F +F(!\"\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/%!G,(*&#\"\"\"\"\"$F(*&% \"xGF(-%$sinG6#,$*&F)F(F+F(F(F(F(F(*&#F(\"\"*F(-%$cosGF.F(F(%\"cGF(" } }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{PARA 0 "" 0 "" {TEXT -1 27 "___________________________" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 27 "__ _________________________" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 " ;" }}}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 3 "Q4 " }}{PARA 0 "" 0 "" {TEXT -1 6 "Find " }{XPPEDIT 18 0 "Int(x*sec^2*x,x);" "6#-%$IntG6$*(% \"xG\"\"\"*$%$secG\"\"#F(F'F(F'" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{SECT 1 {PARA 5 "" 0 "" {TEXT -1 4 "Ans " }}{PARA 0 " " 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "x*tan*x+ln(abs(cos*x))+c;" "6#,( *(%\"xG\"\"\"%$tanGF&F%F&F&-%#lnG6#-%$absG6#*&%$cosGF&F%F&F&%\"cGF&" } {TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 65 "Int(x*sec(x)^2,x);\n``=student[intparts](%,x);\n ``=value(rhs(%))+c;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%$IntG6$*&%\"x G\"\"\")-%$secG6#F'\"\"#F(F'" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/%!G,& *(%\"xG\"\"\"-%$sinG6#F'F(-%$cosGF+!\"\"F(-%$IntG6$*&F)F(F,F.F'F." }} {PARA 11 "" 1 "" {XPPMATH 20 "6#/%!G,(*(%\"xG\"\"\"-%$sinG6#F'F(-%$cos GF+!\"\"F(-%#lnG6#F,F(%\"cGF(" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {PARA 0 "" 0 "" {TEXT -1 46 "More details are shown by using the proce dure " }{TEXT 0 10 "intbyparts" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 72 "Int(x*sec(x) ^2,x);\n``=intbyparts(%,u=x,v,info=true);\n``=value(rhs(%))+c;" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#-%$IntG6$*&%\"xG\"\"\")-%$secG6#F'\"\" #F(F'" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%*PIECEWISEG6$7$/%\"uG%\"xG/ %\"vG*&-%$sinG6#F)\"\"\"-%$cosGF/!\"\"7$/**%\"dGF0F(F0%#~dGF3F)F3F0/** F7F0F+F0F8F3F)F3*$)-%$secGF/\"\"#F0" }}{PARA 11 "" 1 "" {XPPMATH 20 "6 #/-%$IntG6$*&%\"xG\"\"\")-%$secG6#F(\"\"#F)F(-F%6$*&%\"uGF)-%!G6#**%\" dGF)%\"vGF)%#~dG!\"\"F(F:F)F(" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/%!G, &*&%\"uG\"\"\"%\"vGF(F(-%$IntG6$*&F)F(-F$6#**%\"dGF(F'F(%#~dG!\"\"%\"x GF3F(F4F3" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/%!G,&*(%\"xG\"\"\"-%$sin G6#F'F(-%$cosGF+!\"\"F(-%$IntG6$*&F)F(F,F.F'F." }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/%!G,(*(%\"xG\"\"\"-%$sinG6#F'F(-%$cosGF+!\"\"F(-%#lnG6 #F,F(%\"cGF(" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{PARA 0 "" 0 "" {TEXT -1 27 "_____________ ______________" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" } }{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 " " {TEXT -1 27 "___________________________" }}{EXCHG {PARA 0 "> " 0 " " {MPLTEXT 1 0 1 ";" }}}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 3 "Q5 " }} {PARA 0 "" 0 "" {TEXT -1 9 "Find (a) " }{XPPEDIT 18 0 "Int(x*ln*x,x); " "6#-%$IntG6$*(%\"xG\"\"\"%#lnGF(F'F(F'" }{TEXT -1 8 ", (b) " } {XPPEDIT 18 0 "Int(x^2*ln*x,x);" "6#-%$IntG6$*(%\"xG\"\"#%#lnG\"\"\"F' F*F'" }{TEXT -1 8 ", (c) " }{XPPEDIT 18 0 "Int(ln*x/(x^2),x);" "6#-% $IntG6$*(%#lnG\"\"\"%\"xGF(*$F)\"\"#!\"\"F)" }{TEXT -1 7 ", (d) " } {XPPEDIT 18 0 "Int(x^r*ln*x,x);" "6#-%$IntG6$*()%\"xG%\"rG\"\"\"%#lnGF *F(F*F(" }{TEXT -1 8 ", where " }{XPPEDIT 18 0 "r<>-1" "6#0%\"rG,$\"\" \"!\"\"" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{SECT 1 {PARA 5 "" 0 "" {TEXT -1 4 "Ans " }}{PARA 0 "" 0 "" {TEXT -1 4 "(a) " }{XPPEDIT 18 0 "x^2/2" "6#*&%\"xG\"\"#F%!\"\"" }{TEXT -1 1 " " } {XPPEDIT 18 0 "ln*x-x^2/4+c;" "6#,(*&%#lnG\"\"\"%\"xGF&F&*&F'\"\"#\"\" %!\"\"F+%\"cGF&" }{TEXT -1 2 " " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 66 "Int(x*ln(x),x);\n``=student[ intparts](%,ln(x));\n``=value(rhs(%))+c;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%$IntG6$*&%\"xG\"\"\"-%#lnG6#F'F(F'" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/%!G,&*&#\"\"\"\"\"#F(*&)%\"xGF)F(-%#lnG6#F,F(F(F(-%$In tG6$,$*&F)!\"\"F,F(F(F,F5" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/%!G,(*&# \"\"\"\"\"#F(*&)%\"xGF)F(-%#lnG6#F,F(F(F(*&\"\"%!\"\"F,F)F2%\"cGF(" }} }{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 46 "More det ails are shown by using the procedure " }{TEXT 0 10 "intbyparts" } {TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 73 "Int(x*ln(x),x);\n``=intbyparts(%,u=ln(x),v,info= true);\n``=value(rhs(%))+c;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%$IntG 6$*&%\"xG\"\"\"-%#lnG6#F'F(F'" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%*PI ECEWISEG6$7$/%\"uG-%#lnG6#%\"xG/%\"vG,$*&\"\"#!\"\"F,F1\"\"\"7$/**%\"d GF3F(F3%#~dGF2F,F2*&F3F3F,F2/**F7F3F.F3F8F2F,F2F," }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%$IntG6$*&%\"xG\"\"\"-%#lnG6#F(F)F(-F%6$*&%\"uGF)-%!G 6#**%\"dGF)%\"vGF)%#~dG!\"\"F(F8F)F(" }}{PARA 11 "" 1 "" {XPPMATH 20 " 6#/%!G,&*&%\"uG\"\"\"%\"vGF(F(-%$IntG6$*&F)F(-F$6#**%\"dGF(F'F(%#~dG! \"\"%\"xGF3F(F4F3" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/%!G,&*&#\"\"\"\" \"#F(*&)%\"xGF)F(-%#lnG6#F,F(F(F(-%$IntG6$,$*&F)!\"\"F,F(F(F,F5" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#/%!G,(*&#\"\"\"\"\"#F(*&)%\"xGF)F(-%#l nG6#F,F(F(F(*&\"\"%!\"\"F,F)F2%\"cGF(" }}}{PARA 0 "" 0 "" {TEXT -1 0 " " }}{PARA 0 "" 0 "" {TEXT -1 4 "(b) " }{XPPEDIT 18 0 "x^3/3" "6#*&%\"x G\"\"$F%!\"\"" }{TEXT -1 1 " " }{XPPEDIT 18 0 "ln*x-x^3/9+c;" "6#,(*&% #lnG\"\"\"%\"xGF&F&*&F'\"\"$\"\"*!\"\"F+%\"cGF&" }{TEXT -1 1 " " }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 68 "Int(x^2*ln(x),x);\n``=student[intparts](%,ln(x));\n``=value(rhs(%) )+c;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%$IntG6$*&)%\"xG\"\"#\"\"\"-% #lnG6#F(F*F(" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/%!G,&*&#\"\"\"\"\"$F( *&-%#lnG6#%\"xGF()F.F)F(F(F(-%$IntG6$,$*&F)!\"\"F.\"\"#F(F.F5" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#/%!G,(*&#\"\"\"\"\"$F(*&-%#lnG6#%\"xGF ()F.F)F(F(F(*&\"\"*!\"\"F.F)F2%\"cGF(" }}}{PARA 0 "" 0 "" {TEXT -1 0 " " }}{PARA 0 "" 0 "" {TEXT -1 46 "More details are shown by using the p rocedure " }{TEXT 0 10 "intbyparts" }{TEXT -1 2 ". " }}{PARA 0 "" 0 " " {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 75 "Int(x^2*ln (x),x);\n``=intbyparts(%,u=ln(x),v,info=true);\n``=value(rhs(%))+c;" } }{PARA 11 "" 1 "" {XPPMATH 20 "6#-%$IntG6$*&)%\"xG\"\"#\"\"\"-%#lnG6#F (F*F(" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%*PIECEWISEG6$7$/%\"uG-%#lnG 6#%\"xG/%\"vG,$*&\"\"$!\"\"F,F1\"\"\"7$/**%\"dGF3F(F3%#~dGF2F,F2*&F3F3 F,F2/**F7F3F.F3F8F2F,F2*$)F,\"\"#F3" }}{PARA 11 "" 1 "" {XPPMATH 20 "6 #/-%$IntG6$*&)%\"xG\"\"#\"\"\"-%#lnG6#F)F+F)-F%6$*&%\"uGF+-%!G6#**%\"d GF+%\"vGF+%#~dG!\"\"F)F:F+F)" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/%!G,& *&%\"uG\"\"\"%\"vGF(F(-%$IntG6$*&F)F(-F$6#**%\"dGF(F'F(%#~dG!\"\"%\"xG F3F(F4F3" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/%!G,&*&#\"\"\"\"\"$F(*&-% #lnG6#%\"xGF()F.F)F(F(F(-%$IntG6$,$*&F)!\"\"F.\"\"#F(F.F5" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/%!G,&*&#\"\"\"\"\"$F(*&-%#lnG6#%\"xGF()F.F)F(F (F(*&#F(F)F(-%$IntG6$*$)F.\"\"#F(F.F(!\"\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/%!G,(*&#\"\"\"\"\"$F(*&-%#lnG6#%\"xGF()F.F)F(F(F(*&\" \"*!\"\"F.F)F2%\"cGF(" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 5 "(c) " }{XPPEDIT 18 0 "-ln*x/x-1/x+c;" "6#,(*(%#lnG\" \"\"%\"xGF&F'!\"\"F(*&F&F&F'F(F(%\"cGF&" }{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 68 "Int(ln( x)/x^2,x);\n``=student[intparts](%,ln(x));\n``=value(rhs(%))+c;" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#-%$IntG6$*&-%#lnG6#%\"xG\"\"\"F*!\"#F* " }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/%!G,&*&-%#lnG6#%\"xG\"\"\"F*!\"\" F,-%$IntG6$,$*&F+F+*$)F*\"\"#F+F,F,F*F," }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/%!G,(*&-%#lnG6#%\"xG\"\"\"F*!\"\"F,*&F+F+F*F,F,%\"cGF+" }}} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 46 "More deta ils are shown by using the procedure " }{TEXT 0 10 "intbyparts" } {TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 75 "Int(ln(x)/x^2,x);\n``=intbyparts(%,u=ln(x),v,inf o=true);\n``=value(rhs(%))+c;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%$In tG6$*&-%#lnG6#%\"xG\"\"\"F*!\"#F*" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#- %*PIECEWISEG6$7$/%\"uG-%#lnG6#%\"xG/%\"vG,$*&\"\"\"F1F,!\"\"F27$/**%\" dGF1F(F1%#~dGF2F,F2F0/**F6F1F.F1F7F2F,F2*&F1F1*$)F,\"\"#F1F2" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%$IntG6$*&-%#lnG6#%\"xG\"\"\"F+!\"#F+-F%6$ *&%\"uGF,-%!G6#**%\"dGF,%\"vGF,%#~dG!\"\"F+F9F,F+" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/%!G,&*&%\"uG\"\"\"%\"vGF(F(-%$IntG6$*&F)F(-F$6#**%\"dG F(F'F(%#~dG!\"\"%\"xGF3F(F4F3" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/%!G, &*&-%#lnG6#%\"xG\"\"\"F*!\"\"F,-%$IntG6$,$*&F+F+*$)F*\"\"#F+F,F,F*F," }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/%!G,(*&-%#lnG6#%\"xG\"\"\"F*!\"\"F, *&F+F+F*F,F,%\"cGF+" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 " " {TEXT -1 5 "(d) " }{XPPEDIT 18 0 "x^(r+1)/(r+1);" "6#*&)%\"xG,&%\"r G\"\"\"F(F(F(,&F'F(F(F(!\"\"" }{TEXT -1 1 " " }{XPPEDIT 18 0 "ln*x-x^( r+1)/((r+1)^2)+c;" "6#,(*&%#lnG\"\"\"%\"xGF&F&*&)F',&%\"rGF&F&F&F&*$,& F+F&F&F&\"\"#!\"\"F/%\"cGF&" }{TEXT -1 3 ". " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 68 "Int(x^r*ln(x ),x);\n``=student[intparts](%,ln(x));\n``=value(rhs(%))+c;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%$IntG6$*&)%\"xG%\"rG\"\"\"-%#lnG6#F(F*F(" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/%!G,&*(-%#lnG6#%\"xG\"\"\")F*,&%\"r GF+F+F+F+F-!\"\"F+-%$IntG6$*(F*F/F,F+F-F/F*F/" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/%!G,(*(-%#lnG6#%\"xG\"\"\")F*,&%\"rGF+F+F+F+F-!\"\"F+* &F-!\"#F,F+F/%\"cGF+" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 46 "More details are shown by using the procedure " } {TEXT 0 10 "intbyparts" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 75 "Int(x^r*ln(x),x);\n``=i ntbyparts(%,u=ln(x),v,info=true);\n``=value(rhs(%))+c;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%$IntG6$*&)%\"xG%\"rG\"\"\"-%#lnG6#F(F*F(" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#-%*PIECEWISEG6$7$/%\"uG-%#lnG6#%\"xG/% \"vG*&)F,,&%\"rG\"\"\"F3F3F3F1!\"\"7$/**%\"dGF3F(F3%#~dGF4F,F4*&F3F3F, F4/**F8F3F.F3F9F4F,F4)F,F2" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%$IntG 6$*&)%\"xG%\"rG\"\"\"-%#lnG6#F)F+F)-F%6$*&%\"uGF+-%!G6#**%\"dGF+%\"vGF +%#~dG!\"\"F)F:F+F)" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/%!G,&*&%\"uG\" \"\"%\"vGF(F(-%$IntG6$*&F)F(-F$6#**%\"dGF(F'F(%#~dG!\"\"%\"xGF3F(F4F3 " }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/%!G,&*(-%#lnG6#%\"xG\"\"\")F*,&% \"rGF+F+F+F+F-!\"\"F+-%$IntG6$*(F,F+F-F/F*F/F*F/" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/%!G,(*(-%#lnG6#%\"xG\"\"\")F*,&%\"rGF+F+F+F+F-!\"\"F+* &F-!\"#F,F+F/%\"cGF+" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{PARA 0 "" 0 "" {TEXT -1 27 "______ _____________________" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 " " {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" } }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 27 "___________________________" }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{SECT 1 {PARA 4 "" 0 " " {TEXT -1 3 "Q6 " }}{PARA 0 "" 0 "" {TEXT -1 6 "Find " }{XPPEDIT 18 0 "Int(2*x*arctan*x,x);" "6#-%$IntG6$**\"\"#\"\"\"%\"xGF(%'arctanGF(F) F(F)" }{TEXT -1 42 " using the integration by parts formula: " } {XPPEDIT 18 0 "Int(u*``(dv/dx),x) = u*v-Int(v*``(du/dx),x)" "6#/-%$Int G6$*&%\"uG\"\"\"-%!G6#*&%#dvGF)%#dxG!\"\"F)%\"xG,&*&F(F)%\"vGF)F)-F%6$ *&F4F)-F+6#*&%#duGF)F/F0F)F1F0" }{TEXT -1 15 " in two ways: " }} {PARA 0 "" 0 "" {TEXT -1 8 "(i) let " }{XPPEDIT 18 0 "u = arctan*x;" " 6#/%\"uG*&%'arctanG\"\"\"%\"xGF'" }{TEXT -1 5 " and " }{XPPEDIT 18 0 " v=x^2" "6#/%\"vG*$%\"xG\"\"#" }{TEXT -1 12 ", (ii) let " }{XPPEDIT 18 0 "u = arctan*x;" "6#/%\"uG*&%'arctanG\"\"\"%\"xGF'" }{TEXT -1 5 " \+ and " }{XPPEDIT 18 0 "v=1+x^2" "6#/%\"vG,&\"\"\"F&*$%\"xG\"\"#F&" } {TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{SECT 1 {PARA 5 "" 0 "" {TEXT -1 4 "Ans " }}{PARA 0 " " 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "(1+x^2)*arctan*x-x+c;" "6#,(*(,& \"\"\"F&*$%\"xG\"\"#F&F&%'arctanGF&F(F&F&F(!\"\"%\"cGF&" }{TEXT -1 2 " " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 76 "Int(2*x*arctan(x),x);\n``=student[intparts](%,arctan( x));\n``=value(rhs(%))+c;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%$IntG6$ ,$*(\"\"#\"\"\"%\"xGF)-%'arctanG6#F*F)F)F*" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/%!G,&*&)%\"xG\"\"#\"\"\"-%'arctanG6#F(F*F*-%$IntG6$*&, &*$F'F*F*F*F*!\"\"F(F)F(F4" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/%!G,**& )%\"xG\"\"#\"\"\"-%'arctanG6#F(F*F*F(!\"\"F+F*%\"cGF*" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 46 "More details are sho wn by using the procedure " }{TEXT 0 10 "intbyparts" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 83 "Int(2*x*arctan(x),x);\n``=intbyparts(%,u=arctan(x),v,info=true); \n``=value(rhs(%))+c;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%$IntG6$,$*( \"\"#\"\"\"%\"xGF)-%'arctanG6#F*F)F)F*" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%*PIECEWISEG6$7$/%\"uG-%'arctanG6#%\"xG/%\"vG*$)F,\"\"#\"\"\"7$ /**%\"dGF2F(F2%#~dG!\"\"F,F8*&F2F2,&F2F2F/F2F8/**F6F2F.F2F7F8F,F8,$*&F 1F2F,F2F2" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%$IntG6$,$*(\"\"#\"\"\" %\"xGF*-%'arctanG6#F+F*F*F+-F%6$*&%\"uGF*-%!G6#**%\"dGF*%\"vGF*%#~dG! \"\"F+F:F*F+" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/%!G,&*&%\"uG\"\"\"%\" vGF(F(-%$IntG6$*&F)F(-F$6#**%\"dGF(F'F(%#~dG!\"\"%\"xGF3F(F4F3" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#/%!G,&*&-%'arctanG6#%\"xG\"\"\")F*\"\" #F+F+-%$IntG6$*&F*F-,&F+F+*$F,F+F+!\"\"F*F4" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/%!G,**&-%'arctanG6#%\"xG\"\"\")F*\"\"#F+F+F*!\"\"F'F+% \"cGF+" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 6 "(ii) " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 110 "Int(2*x*arctan(x),x);\n``=intbyparts(%,u=arctan(x),v =1+x^2,info=true);\n``=simplify(rhs(%));\n``=value(rhs(%))+c;" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#-%$IntG6$,$*(\"\"#\"\"\"%\"xGF)-%'arct anG6#F*F)F)F*" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%*PIECEWISEG6$7$/%\" uG-%'arctanG6#%\"xG/%\"vG,&\"\"\"F0*$)F,\"\"#F0F07$/**%\"dGF0F(F0%#~dG !\"\"F,F9*&F0F0F/F9/**F7F0F.F0F8F9F,F9,$*&F3F0F,F0F0" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%$IntG6$,$*(\"\"#\"\"\"%\"xGF*-%'arctanG6#F+F*F*F+ -F%6$*&%\"uGF*-%!G6#**%\"dGF*%\"vGF*%#~dG!\"\"F+F:F*F+" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/%!G,&*&%\"uG\"\"\"%\"vGF(F(-%$IntG6$*&F)F(-F$6#** %\"dGF(F'F(%#~dG!\"\"%\"xGF3F(F4F3" }}{PARA 11 "" 1 "" {XPPMATH 20 "6# /%!G,&*&-%'arctanG6#%\"xG\"\"\",&F+F+*$)F*\"\"#F+F+F+F+-%$IntG6$F+F*! \"\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/%!G,(-%'arctanG6#%\"xG\"\"\"* &F&F*)F)\"\"#F*F*-%$IntG6$F*F)!\"\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6 #/%!G,**&-%'arctanG6#%\"xG\"\"\")F*\"\"#F+F+F*!\"\"F'F+%\"cGF+" }}} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{PARA 0 "" 0 "" {TEXT -1 27 "___________________________" }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 27 "__ _________________________" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 " ;" }}}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 3 "Q7 " }}{PARA 0 "" 0 "" {TEXT -1 37 "(a) Use integration by parts to find " }{XPPEDIT 18 0 "In t(arcsin*x,x);" "6#-%$IntG6$*&%'arcsinG\"\"\"%\"xGF(F)" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 54 "(b) Find the area of the region boun ded by the graph " }{XPPEDIT 18 0 "y = arcsin*x;" "6#/%\"yG*&%'arcsin G\"\"\"%\"xGF'" }{TEXT -1 6 ", the " }{TEXT 280 1 "x" }{TEXT -1 19 " a xis and the line " }{XPPEDIT 18 0 "x = sqrt(3)/2;" "6#/%\"xG*&-%%sqrtG 6#\"\"$\"\"\"\"\"#!\"\"" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 54 "(c) Find the area of the region bounded by the graph " }{XPPEDIT 18 0 "y = sin*x;" "6#/%\"yG*&%$sinG\"\"\"%\"xGF'" }{TEXT -1 6 ", the \+ " }{TEXT 281 1 "y" }{TEXT -1 19 " axis and the line " }{XPPEDIT 18 0 " y = sqrt(3)/2;" "6#/%\"yG*&-%%sqrtG6#\"\"$\"\"\"\"\"#!\"\"" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{SECT 1 {PARA 5 "" 0 "" {TEXT -1 3 "Ans" }}{PARA 0 "" 0 "" {TEXT -1 5 "(a) " }{XPPEDIT 18 0 " x*arcsin*x+sqrt(1-x^2)+c;" "6#,(*(%\"xG\"\"\"%'arcsinGF&F%F&F&-%%sqrtG 6#,&F&F&*$F%\"\"#!\"\"F&%\"cGF&" }{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 72 "Int(arcsin(x ),x);\n``=student[intparts](%,arcsin(x));\n``=value(rhs(%))+c;" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#-%$IntG6$-%'arcsinG6#%\"xGF)" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/%!G,&*&-%'arcsinG6#%\"xG\"\"\"F*F+F+-%$IntG 6$*&,&F+F+*$)F*\"\"#F+!\"\"#F4F3F*F+F*F4" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/%!G,(*&-%'arcsinG6#%\"xG\"\"\"F*F+F+*$,&F+F+*$)F*\"\"#F+!\"\"#F +F0F+%\"cGF+" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 46 "More details are shown by using the procedure " }{TEXT 0 10 "intbyparts" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 79 "Int(arcsin(x),x);\n``=intbyp arts(%,u=arcsin(x),v,info=true);\n``=value(rhs(%))+c;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%$IntG6$-%'arcsinG6#%\"xGF)" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%*PIECEWISEG6$7$/%\"uG-%'arcsinG6#%\"xG/%\"vGF,7$/**% \"dG\"\"\"F(F3%#~dG!\"\"F,F5*&F3F3*$,&F3F3*$)F,\"\"#F3F5#F3F;F5/**F2F3 F.F3F4F5F,F5F3" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%$IntG6$-%'arcsinG 6#%\"xGF*-F%6$*&%\"uG\"\"\"-%!G6#**%\"dGF/%\"vGF/%#~dG!\"\"F*F7F/F*" } }{PARA 11 "" 1 "" {XPPMATH 20 "6#/%!G,&*&%\"uG\"\"\"%\"vGF(F(-%$IntG6$ *&F)F(-F$6#**%\"dGF(F'F(%#~dG!\"\"%\"xGF3F(F4F3" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/%!G,&*&-%'arcsinG6#%\"xG\"\"\"F*F+F+-%$IntG6$*&F*F+,&F +F+*$)F*\"\"#F+!\"\"#F4F3F*F4" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/%!G, (*&-%'arcsinG6#%\"xG\"\"\"F*F+F+*$,&F+F+*$)F*\"\"#F+!\"\"#F+F0F+%\"cGF +" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 4 "(b) \+ " }{XPPEDIT 18 0 "sqrt(3)/6*Pi-1/2" "6#,&*(-%%sqrtG6#\"\"$\"\"\"\"\"'! \"\"%#PiGF)F)*&F)F)\"\"#F+F+" }{TEXT -1 2 " " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 189 "p1 := plot( arcsin(x),x=0..1,thickness=2):\np2 := plot(arcsin(x),x=0..sqrt(3)/2,fi lled=true,color=wheat):\np3 := plot([[sqrt(3)/2,0],[sqrt(3)/2,Pi/3]],c olor=black):\nplots[display]([p1,p2,p3]);" }}{PARA 13 "" 1 "" {GLPLOT2D 233 277 277 {PLOTDATA 2 "6'-%'CURVESG6%7U7$$\"\"!F)F(7$$\"3e mmm;arz@!#>$\"3q`%*\\c!)))z@F-7$$\"3[LL$e9ui2%F-$\"3O%omw%QSxSF-7$$\"3 nmmm\"z_\"4iF-$\"3%piP/[\\J@'F-7$$\"3[mmmT&phN)F-$\"39[r%pzCfO)F-7$$\" 3CLLe*=)H\\5!#=$\"3[Y&*f,LB^5FB7$$\"3gmm\"z/3uC\"FB$\"3k%3+D\"fm]7FB7$ $\"3%)***\\7LRDX\"FB$\"3Q-LwFipd9FB7$$\"3]mm\"zR'ok;FB$\"3i-,?nCZs;FB7 $$\"3w***\\i5`h(=FB$\"3%z;47!yL()=FB7$$\"3WLLL3En$4#FB$\"3^qKAX$y#4@FB 7$$\"3qmm;/RE&G#FB$\"3a*pCn[PcI#FB7$$\"3\")*****\\K]4]#FB$\"3G^()HbSyF DFB7$$\"3$******\\PAvr#FB$\"3c1*fEAL@v#FB7$$\"3)******\\nHi#HFB$\"3E_s 5osopHFB7$$\"3jmm\"z*ev:JFB$\"3#>Ie(=r]oJFB7$$\"3?LLL347TLFB$\"3<#3nDF ImS$FB7$$\"3,LLLLY.KNFB$\"3K5[kB1$*4OFB7$$\"3w***\\7o7Tv$FB$\"3!zwsQd/ %[QFB7$$\"3'GLLLQ*o]RFB$\"33Tk9G*G91%FB7$$\"3A++D\"=lj;%FB$\"3UR=4qEU( H%FB7$$\"31++vV&RVzFdgp)\\FB7$$\"3cmm;/T1&*\\FB$\"3%*z\\_6 #*GI_FB7$$\"3&em;zRQb@&FB$\"3]m@'\\!4r'[&FB7$$\"3\\***\\(=>Y2aFB$\"3B2 #\\9GRKr&FB7$$\"39mm;zXu9cFB$\"3[H4N,cmhfFB7$$\"3l******\\y))GeFB$\"3S bp]lPzAiFB7$$\"3'*)***\\i_QQgFB$\"3w](\\u\\zI['FB7$$\"3@***\\7y%3TiFB$ \"3h!erL#***)RnFB7$$\"35****\\P![hY'FB$\"3\\Pl2^GQJqFB7$$\"3kKLL$Qx$om FB$\"3Y%>v^'>d*H(FB7$$\"3!)*****\\P+V)oFB$\"3Yt<%GhAKf(FB7$$\"3?mm\"zp e*zqFB$\"3KWBnjJcmyFB7$$\"3%)*****\\#\\'QH(FB$\"3!QF659ZU<)FB7$$\"3GKL e9S8&\\(FB$\"3wwr\\wrEt%)FB7$$\"3R***\\i?=bq(FB$\"39(4&3nY1(z)FB7$$\"3 \"HLL$3s?6zFB$\"3hI/jM4RE\"*FB7$$\"3a***\\7`Wl7)FB$\"3Aj7*)H$Gp[*FB7$$ \"3#pmmm'*RRL)FB$\"3O!**p^O0A&)*FB7$$\"3Qmm;a<.Y&)FB$\"3W,sYyhyC5!#<7$ $\"3=LLe9tOc()FB$\"3kEqp5Evm5Fcx7$$\"3u******\\Qk\\*)FB$\"3(4j\\nO^$36 Fcx7$$\"3CLL$3dg6<*FB$\"3w:$f5P%yg6Fcx7$$\"3ImmmmxGp$*FB$\"3WR2GAst87F cx7$$\"3A++D\"oK0e*FB$\"3EB*3))\\I,G\"Fcx7$$\"3C+++]oi\"o*FB$\"3%e\"=R zOy<8Fcx7$$\"3A++v=5s#y*FB$\"3m1'*))zk&>O\"Fcx7$$\"35+]P40O\"*)*FB$\"3 /%e9**oeKU\"Fcx7$$\"\"\"F)$\"3c'*[zEjzq:Fcx-%'COLOURG6&%$RGBG$\"#5!\" \"F(F(-%*THICKNESSG6#\"\"#-%)POLYGONSG6U7&7$F(F)7$$\"3wH$>5#*)o()=F-F) 7$Fb\\l$\"3!3D%)y/,y)=F-F'7&Fa\\l7$$\"3yN/ydp:INF-F)7$Fi\\l$\"3HOXM$e! 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" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 53 "Int(ln(x)/x,x);\n``=intbypar ts(%,u=ln(x),v,info=true);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%$IntG6 $*&-%#lnG6#%\"xG\"\"\"F*!\"\"F*" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%* PIECEWISEG6$7$/%\"uG-%#lnG6#%\"xG/%\"vGF)7$/**%\"dG\"\"\"F(F3%#~dG!\" \"F,F5*&F3F3F,F5/**F2F3F.F3F4F5F,F5F6" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%$IntG6$*&-%#lnG6#%\"xG\"\"\"F+!\"\"F+-F%6$*&%\"uGF,-%!G6#**%\"dG F,%\"vGF,%#~dGF-F+F-F,F+" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/%!G,&*&% \"uG\"\"\"%\"vGF(F(-%$IntG6$*&F)F(-F$6#**%\"dGF(F'F(%#~dG!\"\"%\"xGF3F (F4F3" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/%!G,&*$)-%#lnG6#%\"xG\"\"#\" \"\"F--%$IntG6$*&F(F-F+!\"\"F+F2" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 94 "Int(ln(x)/x,x);\n``=student[ changevar](ln(x)=u,%);\n``=value(rhs(%))+c;\n``=subs(u=ln(x),rhs(%)); " }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%$IntG6$*&-%#lnG6#%\"xG\"\"\"F*! \"\"F*" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/%!G-%$IntG6$%\"uGF(" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#/%!G,&*&\"\"#!\"\"%\"uGF'\"\"\"%\"cGF* " }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/%!G,&*&#\"\"\"\"\"#F(*$)-%#lnG6#% \"xGF)F(F(F(%\"cGF(" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {PARA 0 "" 0 "" {TEXT -1 27 "___________________________" }}{PARA 0 " " 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" } }{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 " " {TEXT -1 27 "___________________________" }}{EXCHG {PARA 0 "> " 0 " " {MPLTEXT 1 0 1 ";" }}}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 4 "Q9 " }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 10 " (a) Find " }{XPPEDIT 18 0 "Int(arctan*x/(1+x^2),x);" "6#-%$IntG6$*(%'arctanG\" \"\"%\"xGF(,&F(F(*$F)\"\"#F(!\"\"F)" }{TEXT -1 14 " in two ways: " }} {PARA 0 "" 0 "" {TEXT -1 97 " (i) by using the integration by parts f ormula, (ii) by making a suitable (single) substitution." }}{PARA 0 " " 0 "" {TEXT -1 10 " (b) Find " }{XPPEDIT 18 0 "Int(arctan*x/(x^2),x); " "6#-%$IntG6$*(%'arctanG\"\"\"%\"xGF(*$F)\"\"#!\"\"F)" }{TEXT -1 13 " . Hint: " }{XPPEDIT 18 0 "1/(x*(1+x^2))=1/x-x/(1+x^2)" "6#/*&\"\" \"F%*&%\"xGF%,&F%F%*$F'\"\"#F%F%!\"\",&*&F%F%F'F+F%*&F'F%,&F%F%*$F'F*F %F+F+" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{SECT 1 {PARA 5 "" 0 "" {TEXT -1 3 "Ans" }}{PARA 0 "" 0 "" {TEXT -1 5 "(a) " }{XPPEDIT 18 0 "1/2" "6#*&\"\"\"F$\"\"#!\"\"" }{TEXT -1 1 " " } {XPPEDIT 18 0 "(arctan*x)^2+c;" "6#,&*$*&%'arctanG\"\"\"%\"xGF'\"\"#F' %\"cGF'" }{TEXT -1 2 " " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 80 "Int(arctan(x)/(1+x^2),x);\n``=stude nt[intparts](%,arctan(x));\n``=value(rhs(%))+c;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%$IntG6$*&-%'arctanG6#%\"xG\"\"\",&F+F+*$)F*\"\"#F+F+! \"\"F*" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/%!G,&*$)-%'arctanG6#%\"xG\" \"#\"\"\"F--%$IntG6$*&F(F-,&F-F-*$)F+F,F-F-!\"\"F+F5" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/%!G,&*&#\"\"\"\"\"#F(*$)-%'arctanG6#%\"xGF)F(F(F(% \"cGF(" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 46 "More details are shown by using the procedure " }{TEXT 0 10 "intby parts" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 67 "Int(arctan(x)/(1+x^2),x);\n``=intby parts(%,u=arctan(x),v,info=true);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#- %$IntG6$*&-%'arctanG6#%\"xG\"\"\",&F+F+*$)F*\"\"#F+F+!\"\"F*" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%*PIECEWISEG6$7$/%\"uG-%'arctanG6#%\"xG/%\" vGF)7$/**%\"dG\"\"\"F(F3%#~dG!\"\"F,F5*&F3F3,&F3F3*$)F,\"\"#F3F3F5/**F 2F3F.F3F4F5F,F5F6" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%$IntG6$*&-%'ar ctanG6#%\"xG\"\"\",&F,F,*$)F+\"\"#F,F,!\"\"F+-F%6$*&%\"uGF,-%!G6#**%\" dGF,%\"vGF,%#~dGF1F+F1F,F+" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/%!G,&*& %\"uG\"\"\"%\"vGF(F(-%$IntG6$*&F)F(-F$6#**%\"dGF(F'F(%#~dG!\"\"%\"xGF3 F(F4F3" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/%!G,&*$)-%'arctanG6#%\"xG\" \"#\"\"\"F--%$IntG6$*&F(F-,&F-F-*$)F+F,F-F-!\"\"F+F5" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 112 "Int(arct an(x)/(1+x^2),x);\n``=student[changevar](arctan(x)=u,%);\n``=value(rhs (%))+c;\n``=subs(u=arctan(x),rhs(%));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%$IntG6$*&-%'arctanG6#%\"xG\"\"\",&F+F+*$)F*\"\"#F+F+!\"\"F*" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#/%!G-%$IntG6$%\"uGF(" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/%!G,&*&\"\"#!\"\"%\"uGF'\"\"\"%\"cGF*" }}{PARA 11 " " 1 "" {XPPMATH 20 "6#/%!G,&*&#\"\"\"\"\"#F(*$)-%'arctanG6#%\"xGF)F(F( F(%\"cGF(" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 5 "(b) " }{XPPEDIT 18 0 "-arctan*x/x+ln*x-1/2;" "6#,(*(%'arctanG\" \"\"%\"xGF&F'!\"\"F(*&%#lnGF&F'F&F&*&F&F&\"\"#F(F(" }{TEXT -1 1 " " } {XPPEDIT 18 0 "ln(1+x^2) +c" "6#,&-%#lnG6#,&\"\"\"F(*$%\"xG\"\"#F(F(% \"cGF(" }{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 76 "Int(arctan(x)/x^2,x);\n``=student[i ntparts](%,arctan(x));\n``=value(rhs(%))+c;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%$IntG6$*&-%'arctanG6#%\"xG\"\"\"F*!\"#F*" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/%!G,&*&-%'arctanG6#%\"xG\"\"\"F*!\"\"F,-%$IntG 6$,$*&F+F+*&,&F+F+*$)F*\"\"#F+F+F+F*F+F,F,F*F," }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/%!G,**&-%'arctanG6#%\"xG\"\"\"F*!\"\"F,*&#F+\"\"#F+-%# lnG6#,&F+F+*$)F*F/F+F+F+F,-F1F)F+%\"cGF+" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 46 "More details are shown by using th e procedure " }{TEXT 0 10 "intbyparts" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 83 "Int(arc tan(x)/x^2,x);\n``=intbyparts(%,u=arctan(x),v,info=true);\n``=value(rh s(%))+c;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%$IntG6$*&-%'arctanG6#%\" xG\"\"\"F*!\"#F*" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%*PIECEWISEG6$7$/ %\"uG-%'arctanG6#%\"xG/%\"vG,$*&\"\"\"F1F,!\"\"F27$/**%\"dGF1F(F1%#~dG F2F,F2*&F1F1,&F1F1*$)F,\"\"#F1F1F2/**F6F1F.F1F7F2F,F2*&F1F1*$F;F1F2" } }{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%$IntG6$*&-%'arctanG6#%\"xG\"\"\"F+ !\"#F+-F%6$*&%\"uGF,-%!G6#**%\"dGF,%\"vGF,%#~dG!\"\"F+F9F,F+" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/%!G,&*&%\"uG\"\"\"%\"vGF(F(-%$IntG6$*&F)F(- F$6#**%\"dGF(F'F(%#~dG!\"\"%\"xGF3F(F4F3" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/%!G,&*&-%'arctanG6#%\"xG\"\"\"F*!\"\"F,-%$IntG6$,$*&F+F+*&,&F+F +*$)F*\"\"#F+F+F+F*F+F,F,F*F," }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/%!G, **&-%'arctanG6#%\"xG\"\"\"F*!\"\"F,*&#F+\"\"#F+-%#lnG6#,&F+F+*$)F*F/F+ F+F+F,-F1F)F+%\"cGF+" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {PARA 0 "" 0 "" {TEXT -1 27 "___________________________" }}{PARA 0 " " 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" } }{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 " " {TEXT -1 27 "___________________________" }}{EXCHG {PARA 0 "> " 0 " " {MPLTEXT 1 0 1 ";" }}}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 4 "Q10 " }} {PARA 0 "" 0 "" {TEXT -1 6 "Find " }{XPPEDIT 18 0 "Int(cos*sqrt(x),x) ;" "6#-%$IntG6$*&%$cosG\"\"\"-%%sqrtG6#%\"xGF(F," }{TEXT -1 2 ". " }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{SECT 1 {PARA 5 "" 0 "" {TEXT -1 4 "An s " }}{PARA 0 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "2*sqrt(x)*sin*sqr t(x)+2*cos*sqrt(x)+c" "6#,(**\"\"#\"\"\"-%%sqrtG6#%\"xGF&%$sinGF&-F(6# F*F&F&*(F%F&%$cosGF&-F(6#F*F&F&%\"cGF&" }{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 75 "Int(cos (sqrt(x)),x);\n``=student[intparts](%,2*sqrt(x));\n``=value(rhs(%))+c; " }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%$IntG6$-%$cosG6#*$%\"xG#\"\"\"\" \"#F*" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/%!G,&*(\"\"#\"\"\"%\"xG#F(F' -%$sinG6#*$F)F*F(F(-%$IntG6$*&F)#!\"\"F'F+F(F)F4" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/%!G,(*(\"\"#\"\"\"%\"xG#F(F'-%$sinG6#*$F)F*F(F(*&F'F(- %$cosGF-F(F(%\"cGF(" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 " " {TEXT -1 46 "More details are shown by using the procedure " }{TEXT 0 10 "intbyparts" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 82 "Int(cos(sqrt(x)),x);\n``=int byparts(%,u=2*sqrt(x),v,info=true);\n``=value(rhs(%))+c;" }}{PARA 11 " " 1 "" {XPPMATH 20 "6#-%$IntG6$-%$cosG6#*$%\"xG#\"\"\"\"\"#F*" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#-%*PIECEWISEG6$7$/%\"uG,$*&\"\"#\"\"\" %\"xG#F,F+F,/%\"vG-%$sinG6#*$F-F.7$/**%\"dGF,F(F,%#~dG!\"\"F-F:*&F,F,* $F-#F,F+F:/**F8F,F0F,F9F:F-F:,$*&F.F,*&-%$cosGF3F,F-#F:F+F,F," }} {PARA 11 "" 1 "" {XPPMATH 20 "6#/-%$IntG6$-%$cosG6#*$%\"xG#\"\"\"\"\"# F+-F%6$*&%\"uGF--%!G6#**%\"dGF-%\"vGF-%#~dG!\"\"F+F:F-F+" }}{PARA 11 " " 1 "" {XPPMATH 20 "6#/%!G,&*&%\"uG\"\"\"%\"vGF(F(-%$IntG6$*&F)F(-F$6# **%\"dGF(F'F(%#~dG!\"\"%\"xGF3F(F4F3" }}{PARA 11 "" 1 "" {XPPMATH 20 " 6#/%!G,&*(\"\"#\"\"\"%\"xG#F(F'-%$sinG6#*$F)F*F(F(-%$IntG6$*&F+F(F)#! \"\"F'F)F4" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/%!G,(*(\"\"#\"\"\"%\"xG #F(F'-%$sinG6#*$F)F*F(F(*&F'F(-%$cosGF-F(F(%\"cGF(" }}}{PARA 0 "" 0 " " {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}} {PARA 0 "" 0 "" {TEXT -1 27 "___________________________" }}{PARA 0 " " 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 27 "____________ _______________" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}} {SECT 1 {PARA 4 "" 0 "" {TEXT -1 4 "Q11 " }}{PARA 0 "" 0 "" {TEXT -1 6 "Find " }{XPPEDIT 18 0 "Int((ln*x)^3,x);" "6#-%$IntG6$*$*&%#lnG\"\" \"%\"xGF)\"\"$F*" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 6 "Hint: " }{XPPEDIT 18 0 "Int((ln*x)^2,x)=x*(ln*x)^2-2*x*ln*x+2*x+c" "6#/-%$I ntG6$*$*&%#lnG\"\"\"%\"xGF*\"\"#F+,**&F+F**$*&F)F*F+F*F,F*F***F,F*F+F* F)F*F+F*!\"\"*&F,F*F+F*F*%\"cGF*" }{TEXT -1 17 " from example 4. " }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{SECT 1 {PARA 5 "" 0 "" {TEXT -1 4 "An s " }}{PARA 0 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "x*( ln*x)^3-3*x*( ln*x)^2+6*x*ln*x-6*x+c" "6#,,*&%\"xG\"\"\"*$*&%#lnGF&F%F&\"\"$F&F&*(F* F&F%F&*&F)F&F%F&\"\"#!\"\"**\"\"'F&F%F&F)F&F%F&F&*&F0F&F%F&F.%\"cGF&" }{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 142 "Int(ln(x)^3,x);\n``=student[intparts](%,ln(x)^3 );\n``=student[intparts](rhs(%),ln(x)^2);\n``=student[intparts](rhs(%) ,ln(x));\n``=value(rhs(%))+c;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%$In tG6$*$)-%#lnG6#%\"xG\"\"$\"\"\"F+" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/ %!G,&*&)-%#lnG6#%\"xG\"\"$\"\"\"F+F-F--%$IntG6$,$*&F,F-)F(\"\"#F-F-F+! \"\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/%!G,(*&)-%#lnG6#%\"xG\"\"$\" \"\"F+F-F-*(F,F-)F(\"\"#F-F+F-!\"\"-%$IntG6$,$*&\"\"'F-F(F-F-F+F-" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#/%!G,**&)-%#lnG6#%\"xG\"\"$\"\"\"F+F-F -*(F,F-)F(\"\"#F-F+F-!\"\"*(\"\"'F-F(F-F+F-F--%$IntG6$F3F+F1" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/%!G,,*&)-%#lnG6#%\"xG\"\"$\"\"\"F+F-F-*(F,F -)F(\"\"#F-F+F-!\"\"*(\"\"'F-F(F-F+F-F-*&F3F-F+F-F1%\"cGF-" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 107 "More details o f the first application of the integration by parts formula are shown \+ by using the procedure " }{TEXT 0 10 "intbyparts" }{TEXT -1 2 ". " }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 55 "Int(ln(x)^3,x);\n``=intbyparts(%,u=ln(x)^3,v,info=true);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%$IntG6$*$)-%#lnG6#%\"xG\"\"$\"\"\"F+" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#-%*PIECEWISEG6$7$/%\"uG*$)-%#lnG6#%\"x G\"\"$\"\"\"/%\"vGF.7$/**%\"dGF0F(F0%#~dG!\"\"F.F8,$*(F/F0F+\"\"#F.F8F 0/**F6F0F2F0F7F8F.F8F0" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%$IntG6$*$ )-%#lnG6#%\"xG\"\"$\"\"\"F,-F%6$*&%\"uGF.-%!G6#**%\"dGF.%\"vGF.%#~dG! \"\"F,F:F.F," }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/%!G,&*&%\"uG\"\"\"%\" vGF(F(-%$IntG6$*&F)F(-F$6#**%\"dGF(F'F(%#~dG!\"\"%\"xGF3F(F4F3" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#/%!G,&*&)-%#lnG6#%\"xG\"\"$\"\"\"F+F-F --%$IntG6$,$*&F,F-)F(\"\"#F-F-F+!\"\"" }}}{PARA 0 "" 0 "" {TEXT -1 0 " " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{PARA 0 "" 0 "" {TEXT -1 27 "___________________________" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 27 "___________________________" }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{SECT 1 {PARA 4 "" 0 " " {TEXT -1 5 "Q12 " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 " " {TEXT -1 6 " Find " }{XPPEDIT 18 0 "Int(x^3/sqrt(1-x^2),x);" "6#-%$I ntG6$*&%\"xG\"\"$-%%sqrtG6#,&\"\"\"F-*$F'\"\"#!\"\"F0F'" }{TEXT -1 14 " in two ways: " }}{PARA 0 "" 0 "" {TEXT -1 97 " (i) by initially usi ng the integration by parts formula (ii) by making a suitable substitu tion." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 18 " Hint for (i): Let " }{XPPEDIT 18 0 "u=x^2" "6#/%\"uG*$%\"xG\"\"#" } {TEXT -1 58 ". The residual integral can be found via the substitution " }{XPPEDIT 18 0 "z=1-x^2" "6#/%\"zG,&\"\"\"F&*$%\"xG\"\"#!\"\"" } {TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{SECT 1 {PARA 5 "" 0 "" {TEXT -1 3 "Ans" }}{PARA 0 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "-x^2*sqrt(1-x^2)-2/3" "6#,&*&%\"xG\"\"#-%%sqrtG6#,&\"\"\"F+*$F%F&!\" \"F+F-*&F&F+\"\"$F-F-" }{TEXT -1 1 " " }{XPPEDIT 18 0 "(1-x^2)^(3/2)+c " "6#,&),&\"\"\"F&*$%\"xG\"\"#!\"\"*&\"\"$F&F)F*F&%\"cGF&" }{TEXT -1 6 " or " }{XPPEDIT 18 0 "1/3" "6#*&\"\"\"F$\"\"$!\"\"" }{TEXT -1 1 " " }{XPPEDIT 18 0 "(1-x^2)^(3/2)-sqrt(1-x^2)+c" "6#,(),&\"\"\"F&*$%\"x G\"\"#!\"\"*&\"\"$F&F)F*F&-%%sqrtG6#,&F&F&*$F(F)F*F*%\"cGF&" }{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 4 "(i) " }}{EXCHG {PARA 0 "> " 0 " " {MPLTEXT 1 0 93 "Int(x^3/sqrt(1-x^2),x);\n``=student[intparts](%,x^2 );\n``=simplify(rhs(%));\n``=value(rhs(%))+c;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%$IntG6$*&%\"xG\"\"$,&\"\"\"F**$)F'\"\"#F*!\"\"#F.F-F' " }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/%!G,&*&)%\"xG\"\"#\"\"\",&F*F**$F 'F*!\"\"#F*F)F--%$IntG6$,$*(F)F*F+F.F(F*F-F(F-" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/%!G,&*&)%\"xG\"\"#\"\"\",&F*F**$F'F*!\"\"#F*F)F-*&F)F* -%$IntG6$*&F+F.F(F*F(F*F*" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/%!G,(*&) %\"xG\"\"#\"\"\",&F*F**$F'F*!\"\"#F*F)F-*(F)F*\"\"$F-F+#F0F)F-%\"cGF* " }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 46 "More details are shown by using the procedure " }{TEXT 0 10 "intbyparts" } {TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 100 "Int(x^3/sqrt(1-x^2),x);\n``=intbyparts(%,u=x^2, v,info=true);\n``=simplify(rhs(%));\n``=value(rhs(%))+c;" }}{PARA 11 " " 1 "" {XPPMATH 20 "6#-%$IntG6$*&%\"xG\"\"$,&\"\"\"F**$)F'\"\"#F*!\"\" #F.F-F'" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%*PIECEWISEG6$7$/%\"uG*$)% \"xG\"\"#\"\"\"/%\"vG,$*$,&F-F-F)!\"\"#F-F,F37$/**%\"dGF-F(F-%#~dGF3F+ F3,$*&F,F-F+F-F-/**F8F-F/F-F9F3F+F3*&F+F-F2#F3F," }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%$IntG6$*&%\"xG\"\"$,&\"\"\"F+*$)F(\"\"#F+!\"\"#F/F.F (-F%6$*&%\"uGF+-%!G6#**%\"dGF+%\"vGF+%#~dGF/F(F/F+F(" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/%!G,&*&%\"uG\"\"\"%\"vGF(F(-%$IntG6$*&F)F(-F$6#**% \"dGF(F'F(%#~dG!\"\"%\"xGF3F(F4F3" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/ %!G,&*&)%\"xG\"\"#\"\"\",&F*F**$F'F*!\"\"#F*F)F--%$IntG6$,$*(F)F*F+F.F (F*F-F(F-" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/%!G,&*&)%\"xG\"\"#\"\"\" ,&F*F**$F'F*!\"\"#F*F)F-*&F)F*-%$IntG6$*&F+F.F(F*F(F*F*" }}{PARA 11 " " 1 "" {XPPMATH 20 "6#/%!G,(*&)%\"xG\"\"#\"\"\",&F*F**$F'F*!\"\"#F*F)F -*(F)F*\"\"$F-F+#F0F)F-%\"cGF*" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {PARA 0 "" 0 "" {TEXT -1 5 "(ii) " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 125 "Int(x^3/sqrt(1-x^2),x);\n``=student[changevar](1-x^2 =u,%);\n``=map(expand,rhs(%));\n``=value(rhs(%))+c;\n``=subs(u=1-x^2,r hs(%));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%$IntG6$*&%\"xG\"\"$,&\"\" \"F**$)F'\"\"#F*!\"\"#F.F-F'" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/%!G-% $IntG6$,$*(\"\"#!\"\"%\"uG#F+F*,&\"\"\"F/F,F+F/F+F," }}{PARA 11 "" 1 " " {XPPMATH 20 "6#/%!G-%$IntG6$,&*&\"\"\"F**&\"\"#F*%\"uG#F*F,!\"\"F/*& F,F/F-#F*F,F*F-" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/%!G,(*&\"\"$!\"\"% \"uG#F'\"\"#\"\"\"*$F)#F,F+F(%\"cGF," }}{PARA 11 "" 1 "" {XPPMATH 20 " 6#/%!G,(*&\"\"$!\"\",&\"\"\"F**$)%\"xG\"\"#F*F(#F'F.F**$F)#F*F.F(%\"cG F*" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 " " {TEXT -1 27 "___________________________" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 " " {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" } }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 27 "__ _________________________" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 " ;" }}}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 4 "Q13 " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 9 "Find (a) " }{XPPEDIT 18 0 "Int(exp(2*x)*cos*3*x,x);" "6#-%$IntG6$**-%$expG6#*&\"\"#\"\"\"%\"xGF, F,%$cosGF,\"\"$F,F-F,F-" }{TEXT -1 10 " and (b) " }{XPPEDIT 18 0 "Int (exp(2*x)*sin*3*x,x);" "6#-%$IntG6$**-%$expG6#*&\"\"#\"\"\"%\"xGF,F,%$ sinGF,\"\"$F,F-F,F-" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{SECT 1 {PARA 5 "" 0 "" {TEXT -1 3 "Ans" }}{PARA 0 "" 0 "" {TEXT -1 5 "(a) " }{XPPEDIT 18 0 "2/13" "6#*&\"\"#\"\"\"\"#8!\"\"" }{TEXT -1 1 " " }{XPPEDIT 18 0 "exp(2*x)*cos*3*x+3/13;" "6#,&**-%$expG6#*&\"\"# \"\"\"%\"xGF*F*%$cosGF*\"\"$F*F+F*F**&F-F*\"#8!\"\"F*" }{TEXT -1 1 " \+ " }{XPPEDIT 18 0 "exp(2*x)*sin*3*x+c;" "6#,&**-%$expG6#*&\"\"#\"\"\"% \"xGF*F*%$sinGF*\"\"$F*F+F*F*%\"cGF*" }{TEXT -1 2 " " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 5 "(b) " }{XPPEDIT 18 0 " 2/13" "6#*&\"\"#\"\"\"\"#8!\"\"" }{TEXT -1 1 " " }{XPPEDIT 18 0 "exp(2 *x)*sin*3*x-3/13;" "6#,&**-%$expG6#*&\"\"#\"\"\"%\"xGF*F*%$sinGF*\"\"$ F*F+F*F**&F-F*\"#8!\"\"F0" }{TEXT -1 1 " " }{XPPEDIT 18 0 "exp(2*x)*co s*3*x+c;" "6#,&**-%$expG6#*&\"\"#\"\"\"%\"xGF*F*%$cosGF*\"\"$F*F+F*F*% \"cGF*" }{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT 302 0 "" }{TEXT -1 0 " " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 87 "Int(exp(2*x)*cos(3*x),x) ;\n``=intbyparts(%,u=exp(2*x),v,info=true);\n``=simplify(rhs(%));" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#-%$IntG6$*&-%$expG6#,$*&\"\"#\"\"\"%\" xGF-F-F--%$cosG6#,$*&\"\"$F-F.F-F-F-F." }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%*PIECEWISEG6$7$/%\"uG-%$expG6#,$*&\"\"#\"\"\"%\"xGF/F//%\"vG,$ *&#F/\"\"$F/-%$sinG6#,$*&F6F/F0F/F/F/F/7$/**%\"dGF/F(F/%#~dG!\"\"F0FA, $*&F.F/F)F/F//**F?F/F2F/F@FAF0FA-%$cosGF9" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%$IntG6$*&-%$expG6#,$*&\"\"#\"\"\"%\"xGF.F.F.-%$cosG6 #,$*&\"\"$F.F/F.F.F.F/-F%6$*&%\"uGF.-%!G6#**%\"dGF.%\"vGF.%#~dG!\"\"F/ FAF.F/" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/%!G,&*&%\"uG\"\"\"%\"vGF(F( -%$IntG6$*&F)F(-F$6#**%\"dGF(F'F(%#~dG!\"\"%\"xGF3F(F4F3" }}{PARA 11 " " 1 "" {XPPMATH 20 "6#/%!G,&*&#\"\"\"\"\"$F(*&-%$sinG6#,$*&F)F(%\"xGF( F(F(-%$expG6#,$*&\"\"#F(F0F(F(F(F(F(-%$IntG6$,$*&#F6F)F(F*F(F(F0!\"\" " }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/%!G,&*&#\"\"\"\"\"$F(*&-%$sinG6#, $*&F)F(%\"xGF(F(F(-%$expG6#,$*&\"\"#F(F0F(F(F(F(F(*&#F6F)F(-%$IntG6$F* F0F(!\"\"" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 " " {MPLTEXT 1 0 87 "Int(exp(2*x)*sin(3*x),x);\n``=intbyparts(%,u=exp(2* x),v,info=true);\n``=simplify(rhs(%));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%$IntG6$*&-%$sinG6#,$*&\"\"$\"\"\"%\"xGF-F-F--%$expG6#,$*&\"\"# F-F.F-F-F-F." }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%*PIECEWISEG6$7$/%\"u G-%$expG6#,$*&\"\"#\"\"\"%\"xGF/F//%\"vG,$*&#F/\"\"$F/-%$cosG6#,$*&F6F /F0F/F/F/!\"\"7$/**%\"dGF/F(F/%#~dGF " 0 "" {MPLTEXT 1 0 1 ";" }}}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 27 "____________________ _______" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" } }{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 " " {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" } }}{PARA 0 "" 0 "" {TEXT -1 27 "___________________________" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 4 "Q14 " }}{PARA 0 "" 0 "" {TEXT -1 6 "Find " }{XPPEDIT 18 0 "Int(x^2*arctan(sqrt(x)),x);" "6#-%$IntG6$*&%\"xG\"\"#-%'arctanG6#-% %sqrtG6#F'\"\"\"F'" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT 297 4 "Hi nt" }{TEXT -1 28 ": See example 7. Note that " }{XPPEDIT 18 0 "z^6+1= (z^2+1)*(z^4-z^2+1)" "6#/,&*$%\"zG\"\"'\"\"\"F(F(*&,&*$F&\"\"#F(F(F(F( ,(*$F&\"\"%F(*$F&F,!\"\"F(F(F(" }{TEXT -1 4 " so " }{XPPEDIT 18 0 "z^6 /(1+z^2)=z^4-z^2+1-1/(1+z^2)" "6#/*&%\"zG\"\"',&\"\"\"F(*$F%\"\"#F(!\" \",**$F%\"\"%F(*$F%F*F+F(F(*&F(F(,&F(F(*$F%F*F(F+F+" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{SECT 1 {PARA 5 "" 0 "" {TEXT -1 4 " Ans " }}{PARA 0 "" 0 "" {TEXT -1 2 " " }{XPPEDIT 18 0 "1/3" "6#*&\"\" \"F$\"\"$!\"\"" }{TEXT -1 1 " " }{XPPEDIT 18 0 "x^3*arctan(sqrt(x))-1/ 15" "6#,&*&%\"xG\"\"$-%'arctanG6#-%%sqrtG6#F%\"\"\"F-*&F-F-\"#:!\"\"F0 " }{TEXT -1 1 " " }{XPPEDIT 18 0 "x^(5/2)+1/9" "6#,&)%\"xG*&\"\"&\"\" \"\"\"#!\"\"F(*&F(F(\"\"*F*F(" }{TEXT -1 1 " " }{XPPEDIT 18 0 "x^(3/2) -1/3" "6#,&)%\"xG*&\"\"$\"\"\"\"\"#!\"\"F(*&F(F(F'F*F*" }{TEXT -1 1 " \+ " }{XPPEDIT 18 0 "x^(1/2)+1/3" "6#,&)%\"xG*&\"\"\"F'\"\"#!\"\"F'*&F'F' \"\"$F)F'" }{TEXT -1 1 " " }{XPPEDIT 18 0 "arctan(sqrt(x))+c" "6#,&-%' arctanG6#-%%sqrtG6#%\"xG\"\"\"%\"cGF+" }{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 146 "Int(1/2* x^2*sqrt(x)/(1+x),x);\n``=student[changevar](sqrt(x)=z,%);\n``=map(con vert,rhs(%),parfrac,z);\n``=value(rhs(%))+c;\n``=subs(z=sqrt(x),rhs(%) );" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%$IntG6$,$*(\"\"#!\"\"%\"xG#\" \"&F(,&\"\"\"F.F*F.F)F.F*" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/%!G-%$In tG6$*&%\"zG\"\"',&\"\"\"F,*$)F)\"\"#F,F,!\"\"F)" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/%!G-%$IntG6$,**$)%\"zG\"\"%\"\"\"F-*$)F+\"\"#F-!\"\"F- F-*&F-F-,&F-F-F.F-F1F1F+" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/%!G,,*&\" \"&!\"\"%\"zGF'\"\"\"*&\"\"$F(F)F,F(F)F*-%'arctanG6#F)F(%\"cGF*" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#/%!G,,*&\"\"&!\"\"%\"xG#F'\"\"#\"\"\"* &\"\"$F(F)#F.F+F(*$F)#F,F+F,-%'arctanG6#F0F(%\"cGF," }}}{PARA 0 "" 0 " " {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 88 "Int(x^2*ar ctan(sqrt(x)),x);\n``=student[intparts](%,arctan(sqrt(x)));\n``=value( rhs(%))+c;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%$IntG6$*&-%'arctanG6#* $%\"xG#\"\"\"\"\"#F-)F+F.F-F+" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/%!G, &*&#\"\"\"\"\"$F(*&-%'arctanG6#*$%\"xG#F(\"\"#F()F/F)F(F(F(-%$IntG6$,$ *(\"\"'!\"\"F/#\"\"&F1,&F(F(F/F(F9F(F/F9" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/%!G,.*&#\"\"\"\"\"$F(*&-%'arctanG6#*$%\"xG#F(\"\"#F()F/F)F(F(F( *&\"#:!\"\"F/#\"\"&F1F5*&\"\"*F5F/#F)F1F(*&F)F5F/F0F5*&F'F(F+F(F(%\"cG F(" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 46 "Mo re details are shown by using the procedure " }{TEXT 0 10 "intbyparts " }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 116 "Int(x^2*arctan(sqrt(x)),x);\n``=intbyparts(% ,u=arctan(sqrt(x)),v,info=true);\n``=simplify(rhs(%));\n``=value(rhs(% ))+c;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%$IntG6$*&)%\"xG\"\"#\"\"\"- %'arctanG6#*$F(#F*F)F*F(" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%*PIECEWI SEG6$7$/%\"uG-%'arctanG6#*$%\"xG#\"\"\"\"\"#/%\"vG,$*&\"\"$!\"\"F-F5F/ 7$/**%\"dGF/F(F/%#~dGF6F-F6,$*&F/F/*(F0F/F-#F/F0,&F-F/F/F/F/F6F//**F:F /F2F/F;F6F-F6*$)F-F0F/" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%$IntG6$*& )%\"xG\"\"#\"\"\"-%'arctanG6#*$F)#F+F*F+F)-F%6$*&%\"uGF+-%!G6#**%\"dGF +%\"vGF+%#~dG!\"\"F)F " 0 "" {MPLTEXT 1 0 1 ";" }}}}{PARA 0 "" 0 "" {TEXT -1 27 "___________________________" }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 27 "__ _________________________" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 " ;" }}}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{PARA 0 "" 0 " " {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}} {SECT 1 {PARA 0 "" 0 "" {TEXT -1 18 "Code for pictures " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{SECT 1 {PARA 0 "" 0 "" {TEXT -1 22 "Code for area pictures" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 529 "r3 := evalf(sqrt(3)):\np1 : = plot(arctan(x),x=0..5.5,color=red):\np2 := plot(arctan(x),x=0..r3,fi lled=true,style=patchnogrid,\n color=COLOR(RGB,.85,.85,.85)):\np3 \+ := plot([[0,Pi/2],[5.5,Pi/2]],color=black,linestyle=3):\np4 := plot([[ r3,0],[r3,Pi/3]],color=black):\nt1 := plots[textplot]([[5.5,-.1,`x`],[ -.1,1.9,`y`]],color=black):\nt2 := plots[textplot]([-.15,1.57,`p/2`],c olor=black,font=[SYMBOL,9]):\nplots[display]([p1,p2,p3,p4,t1,t2],label s=[``,``],xtickmarks=[2=`2`,4=`4`],\n ytickmarks=[0=`0`,1=`1`],view=[ -.15..5.5,-.1..1.9]);" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 779 "r3 := evalf(sqrt(3)):\np1 := plot( tan(x),x=0..1.6,y=0..3.3,color=red,discont=true):\nh := evalf(Pi/60): \np2 := plots[polygonplot]([seq([[h*(i-1),tan(h*(i-1))],[h*(i-1),r3], \n [h*i,r3],[h*i,tan(h*i)]],i=1..20)],style=patchnogrid,\n \+ color=COLOR(RGB,.85,.85,.85)):\np3 := plot([[Pi/2,0],[Pi/2,3.3]],colo r=black,linestyle=3):\np4 := plot([[0,r3],[Pi/3,r3]],color=black):\np5 := plot([[Pi/3,r3],[Pi/3,0]],linestyle=2,color=COLOR(RGB,.4,.4,.4)): \nt1 := plots[textplot]([[1.8,-.1,`x`],[-.07,3.3,`y`],\n [-.06,- .1,`O`],[-.06,1.7,`A`],[1.13,1.7,`B`],[1.1,-.1,`C`]],color=black):\nt2 := plots[textplot]([1.57,-.1,`p/2`],color=black,font=[SYMBOL,9]):\npl ots[display]([p1,p2,p3,p4,p5,t1,t2],labels=[``,``],xtickmarks=[1=`1`], \n ytickmarks=[1=`1`,2=`2`,3=`3`],view=[-.1..1.8,-.1..3.3]);" }}} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 357 "r3 := evalf(Pi^2/4):\np1 := plot(sin(sqrt(x)),x=0..4,color=red): \np2 := plot(sin(sqrt(x)),x=0..r3,filled=true,style=patchnogrid,\n \+ color=COLOR(RGB,.85,.85,.85)):\np3 := plot([[r3,0],[r3,1]],color=blac k):\nt1 := plots[textplot]([[4.3,-.05,`x`],[-.1,1.17,`y`]],color=black ):\nplots[display]([p1,p2,p3,t1],labels=[``,``],ytickmarks=3,view=[-.1 5..4.3,-.1..1.17]);" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}}{MARK "4 0 0" 0 }{VIEWOPTS 1 1 0 1 1 1803 1 1 1 1 } {PAGENUMBERS 0 1 2 33 1 1 }