{VERSION 6 0 "IBM INTEL NT" "6.0" } {USTYLETAB {CSTYLE "Maple Input" -1 0 "Courier" 0 1 255 0 0 1 0 1 0 0 1 0 0 0 0 1 }{CSTYLE "2D Math" -1 2 "Times" 0 1 0 0 0 0 0 0 2 0 0 0 0 0 0 1 }{CSTYLE "2D Comment" 2 18 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 1 } {CSTYLE "2D Output" 2 20 "" 0 1 0 0 255 1 0 0 0 0 0 0 0 0 0 1 } {CSTYLE "Blue Emphasis" -1 256 "Times" 0 0 0 0 255 1 0 1 0 0 0 0 0 0 0 1 }{CSTYLE "Green Emphasis" -1 257 "Times" 1 12 0 128 0 1 0 1 0 0 0 0 0 0 0 1 }{CSTYLE "Maroon Emphasis" -1 258 "Times" 1 12 128 0 128 1 0 1 0 0 0 0 0 0 0 1 }{CSTYLE "Purple Emphasis" -1 259 "Times" 1 12 102 0 230 1 0 1 0 0 0 0 0 0 0 0 }{CSTYLE "Red Emphasis" -1 260 "Times " 1 12 255 0 0 1 0 1 0 0 0 0 0 0 0 0 }{CSTYLE "Dark Red Emphasis" -1 261 "Times" 1 12 128 0 0 1 0 1 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 262 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 263 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 264 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 0 }{CSTYLE "" 260 265 "" 1 18 0 0 0 0 0 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 266 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 267 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 268 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 269 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 0 } {CSTYLE "" 261 270 "" 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 0 }{CSTYLE "" -1 271 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 272 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 0 }{CSTYLE "" 260 273 "" 1 18 0 0 0 0 0 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 274 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 0 } {CSTYLE "" -1 275 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 276 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 0 }{PSTYLE "Normal" -1 0 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 "Heading 1" -1 3 1 {CSTYLE "" -1 -1 "Times " 1 18 0 0 128 1 2 1 2 2 2 2 1 1 1 1 }1 1 0 0 8 4 3 0 3 0 2 2 0 1 } {PSTYLE "Heading 2" -1 4 1 {CSTYLE "" -1 -1 "Times" 1 14 128 0 0 1 2 1 2 2 2 2 1 1 1 1 }1 1 0 0 8 2 1 0 1 0 2 2 0 1 }{PSTYLE "Heading 3" -1 5 1 {CSTYLE "" -1 -1 "Times" 1 12 128 0 0 1 1 1 2 2 2 2 1 1 1 1 }1 1 0 0 0 0 1 0 1 0 2 2 0 1 }{PSTYLE "Maple Output" -1 11 1 {CSTYLE "" -1 -1 "Times" 1 12 0 0 0 1 2 2 2 2 2 2 1 1 1 1 }3 3 0 0 0 0 1 0 1 0 2 2 0 1 }{PSTYLE "Normal" -1 256 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 31 "Changing variables in integrals" }}{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 46 "The change of variables formula for integrals " }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 4 "The " }{TEXT 259 27 "change of v ariables formula" }{TEXT -1 19 " for integration is" }}{PARA 256 "" 0 "" {TEXT -1 2 " " }{XPPEDIT 18 0 "Int(f(u)*``(du/dx),x) = Int(f(u),u) ;" "6#/-%$IntG6$*&-%\"fG6#%\"uG\"\"\"-%!G6#*&%#duGF,%#dxG!\"\"F,%\"xG- F%6$-F)6#F+F+" }{TEXT -1 1 "." }}{PARA 256 "" 0 "" {TEXT -1 1 " " } {TEXT 265 14 "______________" }{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 61 "This formula follows from the c hain rule for differentiation." }}{PARA 0 "" 0 "" {TEXT -1 5 "Let " } {XPPEDIT 18 0 "y = Int(f(u),u);" "6#/%\"yG-%$IntG6$-%\"fG6#%\"uGF+" } {TEXT -1 10 ", so that " }{XPPEDIT 18 0 "dy/du = f(u);" "6#/*&%#dyG\" \"\"%#duG!\"\"-%\"fG6#%\"uG" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 13 "Suppose that " }{TEXT 267 1 "u" }{TEXT -1 33 " is a differentia ble function of " }{TEXT 264 1 "x" }{TEXT -1 11 ", so that " } {XPPEDIT 18 0 "dy/dx = dy/du;" "6#/*&%#dyG\"\"\"%#dxG!\"\"*&F%F&%#duGF (" }{TEXT -1 1 " " }{TEXT 263 1 "." }{TEXT -1 1 " " }{XPPEDIT 18 0 "du /dx" "6#*&%#duG\"\"\"%#dxG!\"\"" }{TEXT -1 20 " by the chain rule." } }{PARA 0 "" 0 "" {TEXT -1 6 "Then " }{XPPEDIT 18 0 "f(u);" "6#-%\"fG6 #%\"uG" }{TEXT -1 1 " " }{TEXT 262 1 "." }{TEXT -1 1 " " }{XPPEDIT 18 0 "du/dx = dy/dx;" "6#/*&%#duG\"\"\"%#dxG!\"\"*&%#dyGF&F'F(" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 47 "Integrating this last equatio n with respect to " }{TEXT 268 1 "x" }{TEXT -1 6 " gives" }}{PARA 256 "" 0 "" {TEXT -1 4 " " }{XPPEDIT 18 0 "Int(f(u)*``(du/dx),x) = Int( f(u),u);" "6#/-%$IntG6$*&-%\"fG6#%\"uG\"\"\"-%!G6#*&%#duGF,%#dxG!\"\"F ,%\"xG-F%6$-F)6#F+F+" }{TEXT -1 3 ", " }}{PARA 0 "" 0 "" {TEXT -1 43 " which is the change of variables formula. " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 270 7 "Example" }{TEXT -1 2 ": " }} {PARA 0 "" 0 "" {TEXT -1 25 "We can find the integral " }{XPPEDIT 18 0 "Int((3*x^2+2)^4*x,x);" "6#-%$IntG6$*&,&*&\"\"$\"\"\"*$%\"xG\"\"#F*F *F-F*\"\"%F,F*F," }{TEXT -1 27 " by using the subsitution " } {XPPEDIT 18 0 "u=3*x^2+2" "6#/%\"uG,&*&\"\"$\"\"\"*$%\"xG\"\"#F(F(F+F( " }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 6 "Given " }{XPPEDIT 18 0 "u=3*x^2+2" "6#/%\"uG,&*&\"\"$\"\"\"*$%\"xG\"\"#F(F(F+F(" }{TEXT -1 19 ", it follows that " }{XPPEDIT 18 0 "du/dx = 6*x;" "6#/*&%#duG\"\" \"%#dxG!\"\"*&\"\"'F&%\"xGF&" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 20 "Then the expression " }{XPPEDIT 18 0 "(3*x^2+2)^4" "6#*$, &*&\"\"$\"\"\"*$%\"xG\"\"#F'F'F*F'\"\"%" }{TEXT -1 30 " in the integra l has the form " }{XPPEDIT 18 0 "f(u)=u^4" "6#/-%\"fG6#%\"uG*$F'\"\"% " }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 6 "Now " }{XPPEDIT 18 0 "Int(f(u)*``(du/dx),x) = Int(u^4*`.`*6*x,x)" "6#/-%$IntG6$*&-%\"fG6# %\"uG\"\"\"-%!G6#*&%#duGF,%#dxG!\"\"F,%\"xG-F%6$**F+\"\"%%\".GF,\"\"'F ,F4F,F4" }{XPPEDIT 18 0 " ``=Int((3*x^2+2)^4*6*x,x)" "6#/%!G-%$IntG6$* (,&*&\"\"$\"\"\"*$%\"xG\"\"#F,F,F/F,\"\"%\"\"'F,F.F,F." }{TEXT -1 85 " , which is not exactly the same as the given integral because of the e xtra factor 6. " }}{PARA 0 "" 0 "" {TEXT -1 7 "However" }}{PARA 256 " " 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "Int((3*x^2+2)^4*x,x) = 1/6;" "6# /-%$IntG6$*&,&*&\"\"$\"\"\"*$%\"xG\"\"#F+F+F.F+\"\"%F-F+F-*&F+F+\"\"'! \"\"" }{TEXT -1 1 " " }{XPPEDIT 18 0 "Int((3*x^2+2)^4*6*x,x) = 1/6;" " 6#/-%$IntG6$*(,&*&\"\"$\"\"\"*$%\"xG\"\"#F+F+F.F+\"\"%\"\"'F+F-F+F-*&F +F+F0!\"\"" }{TEXT -1 2 " " }{XPPEDIT 18 0 "Int(u^4*``(du/dx),x)" "6# -%$IntG6$*&%\"uG\"\"%-%!G6#*&%#duG\"\"\"%#dxG!\"\"F.%\"xG" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 80 "Applying the change of variables formula the last expression can be written as: " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "1/6" "6#*&\"\"\"F$\"\"'!\"\"" }{TEXT -1 1 " " }{XPPEDIT 18 0 "Int(u^4,u)" "6#-%$IntG6$*$%\"uG\"\"%F'" } {TEXT -1 1 " " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "``=1 /6" "6#/%!G*&\"\"\"F&\"\"'!\"\"" }{TEXT -1 1 " " }{XPPEDIT 18 0 "``(u^ 5/5)+c;" "6#,&-%!G6#*&%\"uG\"\"&F)!\"\"\"\"\"%\"cGF+" }{TEXT -1 1 " " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "``=u^5/30 + c" "6#/ %!G,&*&%\"uG\"\"&\"#I!\"\"\"\"\"%\"cGF+" }{TEXT -1 2 ". " }}{PARA 0 " " 0 "" {TEXT -1 6 "Hence " }}{PARA 256 "" 0 "" {TEXT -1 1 " " } {XPPEDIT 18 0 "Int((3*x^2+2)^4*x,x) = (3*x^2+2)^5/30+c" "6#/-%$IntG6$* &,&*&\"\"$\"\"\"*$%\"xG\"\"#F+F+F.F+\"\"%F-F+F-,&*&,&*&F*F+*$F-F.F+F+F .F+\"\"&\"#I!\"\"F+%\"cGF+" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 68 "This result can be checked by d ifferentiation using the chain rule. " }}{PARA 0 "" 0 "" {TEXT -1 54 " Doing this the \"long way\", by making the substitution " }{XPPEDIT 18 0 "u=3*x^2+2" "6#/%\"uG,&*&\"\"$\"\"\"*$%\"xG\"\"#F(F(F+F(" }{TEXT -1 90 ", shows how the application of the change of variables formula \+ \"reverses the chain rule\". " }}{PARA 0 "" 0 "" {TEXT -1 5 "Let " } {XPPEDIT 18 0 "y=(3*x^2+2)^5/30" "6#/%\"yG*&,&*&\"\"$\"\"\"*$%\"xG\"\" #F)F)F,F)\"\"&\"#I!\"\"" }{TEXT -1 9 " so that " }{XPPEDIT 18 0 "y=u^5 /30" "6#/%\"yG*&%\"uG\"\"&\"#I!\"\"" }{TEXT -1 7 " and " }{XPPEDIT 18 0 "dy/du=u^4/6" "6#/*&%#dyG\"\"\"%#duG!\"\"*&%\"uG\"\"%\"\"'F(" } {TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 12 "Then, since " } {XPPEDIT 18 0 "du/dx=6*x" "6#/*&%#duG\"\"\"%#dxG!\"\"*&\"\"'F&%\"xGF& " }{TEXT -1 24 ", the chain rule gives " }{XPPEDIT 18 0 "dy/dx=dy/du " "6#/*&%#dyG\"\"\"%#dxG!\"\"*&F%F&%#duGF(" }{TEXT -1 1 " " }{TEXT 271 1 "." }{TEXT -1 1 " " }{XPPEDIT 18 0 "du/dx = u^4/6" "6#/*&%#duG\" \"\"%#dxG!\"\"*&%\"uG\"\"%\"\"'F(" }{TEXT -1 1 " " }{TEXT 272 1 "." } {TEXT -1 1 " " }{XPPEDIT 18 0 "6*x = u^4*`.`*x" "6#/*&\"\"'\"\"\"%\"xG F&*(%\"uG\"\"%%\".GF&F'F&" }{XPPEDIT 18 0 "`` = (3*x^2+2)^4*x" "6#/%!G *&,&*&\"\"$\"\"\"*$%\"xG\"\"#F)F)F,F)\"\"%F+F)" }{TEXT -1 37 ", which \+ is the original integrand. " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {PARA 0 "" 0 "" {TEXT -1 3 " " }}{PARA 0 "" 0 "" {TEXT -1 54 "The sw itch to the new integral involving the variable " }{TEXT 269 1 "u" } {TEXT -1 43 " can be achieved by rewriting the equation " }{XPPEDIT 18 0 "du/dx=6*x" "6#/*&%#duG\"\"\"%#dxG!\"\"*&\"\"'F&%\"xGF&" }{TEXT -1 8 " in the " }{TEXT 259 13 "symbolic form" }{TEXT -1 1 " " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "du = 6*x*dx;" "6#/%#duG*(\" \"'\"\"\"%\"xGF'%#dxGF'" }{TEXT -1 2 ", " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{TEXT 273 6 "______" }{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 10 "or even " }{XPPEDIT 18 0 "1/6;" "6#*&\"\"\"F$\"\"'!\"\" " }{TEXT -1 1 " " }{TEXT 266 1 "." }{TEXT -1 1 " " }{XPPEDIT 18 0 "du \+ = x*dx;" "6#/%#duG*&%\"xG\"\"\"%#dxGF'" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 13 "We may write:" }}{PARA 256 "" 0 "" {TEXT -1 1 " " } {XPPEDIT 18 0 "Int((3*x^2+2)^4*x,x);" "6#-%$IntG6$*&,&*&\"\"$\"\"\"*$% \"xG\"\"#F*F*F-F*\"\"%F,F*F," }{TEXT -1 21 " --- " } {XPPEDIT 18 0 "PIECEWISE([u = 3*x^2+2, ``],[du = 6*x*dx, ``],[` `*``(1 /6)*`.`*du = x*dx, ``]);" "6#-%*PIECEWISEG6%7$/%\"uG,&*&\"\"$\"\"\"*$% \"xG\"\"#F,F,F/F,%!G7$/%#duG*(\"\"'F,F.F,%#dxGF,F07$/**%\"~GF,-F06#*&F ,F,F5!\"\"F,%\".GF,F3F,*&F.F,F6F,F0" }{TEXT -1 2 " " }}{PARA 256 "" 0 "" {TEXT -1 3 " = " }{XPPEDIT 18 0 "Int(u^4*`.`*``(1/6)*`.`,u);" "6# -%$IntG6$**%\"uG\"\"%%\".G\"\"\"-%!G6#*&F*F*\"\"'!\"\"F*F)F*F'" } {TEXT -1 3 ". " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "`` = 1/6" "6#/%!G*&\"\"\"F&\"\"'!\"\"" }{TEXT -1 1 " " }{XPPEDIT 18 0 "In t(u^4,u)" "6#-%$IntG6$*$%\"uG\"\"%F'" }{TEXT -1 1 " " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "``=u^5/30+c" "6#/%!G,&*&%\"uG\"\"& \"#I!\"\"\"\"\"%\"cGF+" }{TEXT -1 1 " " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "``=(3*x^2+2)^5/30+c" "6#/%!G,&*&,&*&\"\"$\"\"\"* $%\"xG\"\"#F*F*F-F*\"\"&\"#I!\"\"F*%\"cGF*" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 259 4 "Note" }{TEXT -1 94 ": By using this \"symbolic device\" we avoid having to mention \+ the change of variables formula. " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {PARA 0 "" 0 "" {TEXT -1 4 "The " }{TEXT 0 7 "student" }{TEXT -1 30 " \+ package contains a procedure " }{TEXT 0 9 "changevar" }{TEXT -1 101 " \+ which allows the variable in an integral to be changed according to th e change of variables formula." }}{PARA 0 "" 0 "" {TEXT -1 26 "For exa mple, the integral " }{XPPEDIT 18 0 "Int((3*x^2+2)^4*x,x);" "6#-%$IntG 6$*&,&*&\"\"$\"\"\"*$%\"xG\"\"#F*F*F-F*\"\"%F,F*F," }{TEXT -1 46 " can be simplified by making the substitution " }{XPPEDIT 18 0 "u = 3*x^2+ 2;" "6#/%\"uG,&*&\"\"$\"\"\"*$%\"xG\"\"#F(F(F+F(" }{TEXT -1 1 "." }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 53 "student[changevar](u=3*x^2+2,Int((3*x^2+2)^4*x,x),u);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%$IntG6$,$*&\"\"'!\"\"%\"uG\"\"%\"\"\"F*" }}} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 108 "The comp lete process of finding the integral by an explicit change of variable s can be achieved as follows. " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 82 "student[changevar](u=3*x^2+2 ,Int((3*x^2+2)^4*x,x),u);\nvalue(%);\nsubs(u=3*x^2+2,%);" }}{PARA 11 " " 1 "" {XPPMATH 20 "6#-%$IntG6$,$*&\"\"'!\"\"%\"uG\"\"%\"\"\"F*" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#,$*&\"#I!\"\"%\"uG\"\"&\"\"\"" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#,$*&\"#I!\"\",&*&\"\"$\"\"\")%\"xG\"\" #F*F*F-F*\"\"&F*" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 259 2 "OR" }{TEXT -1 2 ": " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 119 "Int((3*x^2+2)^4*x,x);\nsubstitution := u=3*x^2+2;\nstudent[ch angevar](substitution,%%,u);\nvalue(%);\nsubs(substitution,%);" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#-%$IntG6$*&),&*&\"\"$\"\"\")%\"xG\"\"# F+F+F.F+\"\"%F+F-F+F-" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%-substituti onG/%\"uG,&*&\"\"$\"\"\")%\"xG\"\"#F*F*F-F*" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%$IntG6$,$*&\"\"'!\"\"%\"uG\"\"%\"\"\"F*" }}{PARA 11 " " 1 "" {XPPMATH 20 "6#,$*&\"#I!\"\"%\"uG\"\"&\"\"\"" }}{PARA 11 "" 1 " " {XPPMATH 20 "6#,$*&\"#I!\"\",&*&\"\"$\"\"\")%\"xG\"\"#F*F*F-F*\"\"&F *" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {PARA 0 "" 0 "" {TEXT -1 119 "If we get Maple to find the integral dir ectly, we obtain an expanded form for the result which differs by the \+ constant " }{XPPEDIT 18 0 "32/30=16/15" "6#/*&\"#K\"\"\"\"#I!\"\"*&\"# ;F&\"#:F(" }{TEXT -1 27 " from the previous result. " }}{PARA 0 "" 0 " " {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 51 "Int((3*x^2 +2)^4*x,x);\nvalue(%);\n%+16/15;\nfactor(%);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%$IntG6$*&),&*&\"\"$\"\"\")%\"xG\"\"#F+F+F.F+\"\"%F+F- F+F-" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,,*&#\"#\")\"#5\"\"\"*$)%\"xGF 'F(F(F(*&\"#FF()F+\"\")F(F(*&\"#OF()F+\"\"'F(F(*&\"#CF()F+\"\"%F(F(*&F /F()F+\"\"#F(F(" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,.*&#\"#\")\"#5\"\" \"*$)%\"xGF'F(F(F(*&\"#FF()F+\"\")F(F(*&\"#OF()F+\"\"'F(F(*&\"#CF()F+ \"\"%F(F(*&F/F()F+\"\"#F(F(#\"#;\"#:F(" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,$*&\"#I!\"\",&*&\"\"$\"\"\")%\"xG\"\"#F*F*F-F*\"\"&F*" }}} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 259 4 "Note" }{TEXT -1 67 ": We can get Maple to come \+ up with the change of variables formula." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 91 "f := 'f': u := 'u': \+ \nInt(f(u),u);\nstudent[changevar](u=g(x),Int(f(u),u),x);\nsubs(g(x)=u ,%);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%$IntG6$-%\"fG6#%\"uGF)" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#-%$IntG6$*&-%\"fG6#-%\"gG6#%\"xG\"\"\" -%%diffG6$F*F-F.F-" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%$IntG6$*&-%\"f G6#%\"uG\"\"\"-%%diffG6$F*%\"xGF+F/" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 60 "Examples of finding integrals by the method of subst itution " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 10 "Example 1 " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 13 "The integral " }{XPPEDIT 18 0 "Int(x ^2*(1+x^3)^5,x)" "6#-%$IntG6$*&%\"xG\"\"#,&\"\"\"F**$F'\"\"$F*\"\"&F' " }{TEXT -1 41 " can be found by using the substitution " }{XPPEDIT 18 0 "u = 1+x^3;" "6#/%\"uG,&\"\"\"F&*$%\"xG\"\"$F&" }{TEXT -1 11 ", \+ so that " }{TEXT 274 1 " " }{XPPEDIT 18 0 "du/dx = 3*x^2;" "6#/*&%#duG \"\"\"%#dxG!\"\"*&\"\"$F&*$%\"xG\"\"#F&" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 256 "" 0 "" {TEXT -1 24 " \+ " }{XPPEDIT 18 0 "Int(x^2*(1+x^3)^5,x);" "6#-%$IntG6$*&%\"xG\"\" #,&\"\"\"F**$F'\"\"$F*\"\"&F'" }{TEXT -1 14 " --- " } {XPPEDIT 18 0 "PIECEWISE([u = 1+x^3, ``],[``, ``],[du = 3*x^2*dx, ``(1 /3)*`.`*du = x^2*dx]);" "6#-%*PIECEWISEG6%7$/%\"uG,&\"\"\"F**$%\"xG\" \"$F*%!G7$F.F.7$/%#duG*(F-F**$F,\"\"#F*%#dxGF*/*(-F.6#*&F*F*F-!\"\"F*% \".GF*F2F**&F,F5F6F*" }{TEXT -1 2 " " }}{PARA 0 "" 0 "" {TEXT -1 0 " " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "``=1/3" "6#/%!G*& \"\"\"F&\"\"$!\"\"" }{TEXT -1 1 " " }{XPPEDIT 18 0 "Int(u^5,u);" "6#-% $IntG6$*$%\"uG\"\"&F'" }{TEXT -1 1 " " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "`` = 1/3;" "6#/%!G*&\"\"\"F&\"\"$!\"\"" }{TEXT -1 1 " " }{XPPEDIT 18 0 "``(u^6/6)+c" "6#,&-%!G6#*&%\"uG\"\"'F)!\"\"\"\" \"%\"cGF+" }{TEXT -1 4 " " }}{PARA 256 "" 0 "" {TEXT -1 1 " " } {XPPEDIT 18 0 "`` = u^6/18+c;" "6#/%!G,&*&%\"uG\"\"'\"#=!\"\"\"\"\"%\" cGF+" }{TEXT -1 9 " " }}{PARA 256 "" 0 "" {TEXT -1 1 " " } {XPPEDIT 18 0 "`` = 1/18;" "6#/%!G*&\"\"\"F&\"#=!\"\"" }{TEXT -1 1 " \+ " }{XPPEDIT 18 0 "(1+x^3)^6+c;" "6#,&*$,&\"\"\"F&*$%\"xG\"\"$F&\"\"'F& %\"cGF&" }{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 117 "Int(x^ 2*(1+x^3)^5,x);\nsubstitution := u=1+x^3;\nstudent[changevar](substitu tion,%%,u);\nvalue(%);\nsubs(substitution,%);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%$IntG6$*&)%\"xG\"\"#\"\"\"),&F*F**$)F(\"\"$F*F*\"\"&F *F(" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%-substitutionG/%\"uG,&\"\"\"F (*$)%\"xG\"\"$F(F(" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%$IntG6$,$*&\" \"$!\"\"%\"uG\"\"&\"\"\"F*" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,$*&\"#= !\"\"%\"uG\"\"'\"\"\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,$*&\"#=!\"\" ,&\"\"\"F(*$)%\"xG\"\"$F(F(\"\"'F(" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" } }{PARA 0 "" 0 "" {TEXT -1 119 "If we get Maple to find the integral di rectly, we obtain an expanded form for the result which differs by the constant " }{XPPEDIT 18 0 "1/18;" "6#*&\"\"\"F$\"#=!\"\"" }{TEXT -1 26 " from the previous result." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 50 "Int(x^2*(1+x^3)^5,x);\nvalue (%);\n%+1/18;\nfactor(%);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%$IntG6$ *&)%\"xG\"\"#\"\"\"),&F*F**$)F(\"\"$F*F*\"\"&F*F(" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,.*&#\"\"\"\"#=F&*$)%\"xGF'F&F&F&*&#F&\"\"$F&*$)F*\"#:F &F&F&*&#\"\"&\"\"'F&*$)F*\"#7F&F&F&*&#\"#5\"\"*F&*$)F*F;F&F&F&*&F2F&*$ )F*F4F&F&F&*&F,F&*$)F*F-F&F&F&" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,0*& #\"\"\"\"#=F&*$)%\"xGF'F&F&F&*&#F&\"\"$F&*$)F*\"#:F&F&F&*&#\"\"&\"\"'F &*$)F*\"#7F&F&F&*&#\"#5\"\"*F&*$)F*F;F&F&F&*&F2F&*$)F*F4F&F&F&*&F,F&*$ )F*F-F&F&F&F%F&" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,$*(\"#=!\"\",&%\"x G\"\"\"F)F)\"\"',(*$)F(\"\"#F)F)F(F&F)F)F*F)" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 64 "To see that this is the s ame as the previous result, note that " }{XPPEDIT 18 0 "1+x^3=(x+1)*( x^2-x+1)" "6#/,&\"\"\"F%*$%\"xG\"\"$F%*&,&F'F%F%F%F%,(*$F'\"\"#F%F'!\" \"F%F%F%" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 10 "Example 2 " }}{PARA 256 "" 0 "" {TEXT -1 24 " \+ " }{XPPEDIT 18 0 "Int(x^3*cos(x^4),x);" "6#-%$IntG6$*&%\"x G\"\"$-%$cosG6#*$F'\"\"%\"\"\"F'" }{TEXT -1 14 " --- " } {XPPEDIT 18 0 "PIECEWISE([u = x^4, ``],[``, ``],[du = 4*x^3*dx, ``(1/4 )*`.`*du = x^3*dx]);" "6#-%*PIECEWISEG6%7$/%\"uG*$%\"xG\"\"%%!G7$F,F,7 $/%#duG*(F+\"\"\"*$F*\"\"$F2%#dxGF2/*(-F,6#*&F2F2F+!\"\"F2%\".GF2F0F2* &F*F4F5F2" }{TEXT -1 2 " " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "`` = 1/4;" "6#/%!G*&\"\"\"F &\"\"%!\"\"" }{TEXT -1 1 " " }{XPPEDIT 18 0 "Int(cos*u,u);" "6#-%$IntG 6$*&%$cosG\"\"\"%\"uGF(F)" }{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "`` = 1/4;" "6# /%!G*&\"\"\"F&\"\"%!\"\"" }{TEXT -1 1 " " }{XPPEDIT 18 0 "sin*u+c;" "6 #,&*&%$sinG\"\"\"%\"uGF&F&%\"cGF&" }{TEXT -1 4 " " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "`` = 1/4;" "6#/%!G*&\"\"\"F&\"\"%! \"\"" }{TEXT -1 1 " " }{XPPEDIT 18 0 "sin(x^4)+c;" "6#,&-%$sinG6#*$%\" xG\"\"%\"\"\"%\"cGF*" }{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 114 "Int(x^3*cos(x^4),x);\nsub stitution := u=x^4;\nstudent[changevar](substitution,%%,u);\nvalue(%); \nsubs(substitution,%);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%$IntG6$*& )%\"xG\"\"$\"\"\"-%$cosG6#*$)F(\"\"%F*F*F(" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%-substitutionG/%\"uG*$)%\"xG\"\"%\"\"\"" }}{PARA 11 " " 1 "" {XPPMATH 20 "6#-%$IntG6$,$*&#\"\"\"\"\"%F)-%$cosG6#%\"uGF)F)F. " }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,$*&#\"\"\"\"\"%F&-%$sinG6#%\"uGF& F&" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,$*&#\"\"\"\"\"%F&-%$sinG6#*$)% \"xGF'F&F&F&" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 43 "Maple can also find the integral directly. " }}{PARA 0 " " 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 30 "Int(x ^3*cos(x^4),x);\nvalue(%);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%$IntG6 $*&)%\"xG\"\"$\"\"\"-%$cosG6#*$)F(\"\"%F*F*F(" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,$*&#\"\"\"\"\"%F&-%$sinG6#*$)%\"xGF'F&F&F&" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 10 "Example 3 " }}{PARA 256 "" 0 "" {TEXT -1 24 " " }{XPPEDIT 18 0 "Int((2*x-3)* (x^2-3*x+1)^3,x);" "6#-%$IntG6$*&,&*&\"\"#\"\"\"%\"xGF*F*\"\"$!\"\"F** $,(*$F+F)F**&F,F*F+F*F-F*F*F,F*F+" }{TEXT -1 14 " --- " } {XPPEDIT 18 0 "PIECEWISE([u = x^2-3*x+1, ``],[``, ``],[du = (2*x-3)*dx , ``]);" "6#-%*PIECEWISEG6%7$/%\"uG,(*$%\"xG\"\"#\"\"\"*&\"\"$F-F+F-! \"\"F-F-%!G7$F1F17$/%#duG*&,&*&F,F-F+F-F-F/F0F-%#dxGF-F1" }{TEXT -1 2 " " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "``=Int(u^3,u) " "6#/%!G-%$IntG6$*$%\"uG\"\"$F)" }{TEXT -1 1 " " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "``=u^4/4+c" "6#/%!G,&*&%\"uG\"\"%F(!\" \"\"\"\"%\"cGF*" }{TEXT -1 1 " " }}{PARA 256 "" 0 "" {TEXT -1 1 " " } {XPPEDIT 18 0 "``=1/4" "6#/%!G*&\"\"\"F&\"\"%!\"\"" }{TEXT -1 1 " " } {XPPEDIT 18 0 "(x^2-3*x+1)^4+c;" "6#,&*$,(*$%\"xG\"\"#\"\"\"*&\"\"$F)F 'F)!\"\"F)F)\"\"%F)%\"cGF)" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 129 "Int((2*x-3)*(x^2- 3*x+1)^3,x);\nsubstitution := u=x^2-3*x+1;\nstudent[changevar](substit ution,%%,u);\nvalue(%);\nsubs(substitution,%);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%$IntG6$*&,&*&\"\"#\"\"\"%\"xGF*F*\"\"$!\"\"F*),(*$)F+ F)F*F**&F,F*F+F*F-F*F*F,F*F+" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%-sub stitutionG/%\"uG,(*$)%\"xG\"\"#\"\"\"F,*&\"\"$F,F*F,!\"\"F,F," }} {PARA 11 "" 1 "" {XPPMATH 20 "6#-%$IntG6$*$)%\"uG\"\"$\"\"\"F(" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#,$*&\"\"%!\"\"%\"uGF%\"\"\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,$*&\"\"%!\"\",(*$)%\"xG\"\"#\"\"\"F,*&\"\"$ F,F*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 10 "Ex ample 4 " }}{PARA 256 "" 0 "" {TEXT -1 24 " " } {XPPEDIT 18 0 "Int(x*sqrt(1-x^2),x);" "6#-%$IntG6$*&%\"xG\"\"\"-%%sqrt G6#,&F(F(*$F'\"\"#!\"\"F(F'" }{TEXT -1 14 " --- " }{XPPEDIT 18 0 "PIECEWISE([u = 1-x^2, ``],[``, ``],[du = -2*x*dx, ``(-1/2)*`.`*d u = x*dx]);" "6#-%*PIECEWISEG6%7$/%\"uG,&\"\"\"F**$%\"xG\"\"#!\"\"%!G7 $F/F/7$/%#duG,$*(F-F*F,F*%#dxGF*F./*(-F/6#,$*&F*F*F-F.F.F*%\".GF*F3F** &F,F*F6F*" }{TEXT -1 2 " " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "``=-1/2" "6#/%!G,$*&\"\"\"F '\"\"#!\"\"F)" }{TEXT -1 1 " " }{XPPEDIT 18 0 "Int(sqrt(u),u);" "6#-%$ IntG6$-%%sqrtG6#%\"uGF)" }{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "``=-1/2" "6#/%!G ,$*&\"\"\"F'\"\"#!\"\"F)" }{TEXT -1 1 " " }{XPPEDIT 18 0 "``(2/3*u^(3/ 2))+c" "6#,&-%!G6#*(\"\"#\"\"\"\"\"$!\"\")%\"uG*&F*F)F(F+F)F)%\"cGF)" }{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 256 "" 0 "" {XPPEDIT 18 0 "``=-1/3" "6#/%!G,$*&\"\"\"F'\"\"$!\"\"F)" }{TEXT -1 1 " " }{XPPEDIT 18 0 "u^(3/2)+c;" "6#,&)%\"uG*&\"\"$\"\"\"\"\"#!\"\"F(%\" cGF(" }{TEXT -1 1 " " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "`` = -1/3;" "6#/%!G,$*&\"\"\"F'\"\"$!\"\"F)" }{TEXT -1 1 " " } {XPPEDIT 18 0 "(1-x^2)^(3/2)+c;" "6#,&),&\"\"\"F&*$%\"xG\"\"#!\"\"*&\" \"$F&F)F*F&%\"cGF&" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 0 "" } }{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 117 "Int(x*sqrt(1-x^2),x);\nsub stitution := u=1-x^2;\nstudent[changevar](substitution,%%,u);\nvalue(% );\nsubs(substitution,%);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%$IntG6$ *&%\"xG\"\"\",&F(F(*$)F'\"\"#F(!\"\"#F(F,F'" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%-substitutionG/%\"uG,&\"\"\"F(*$)%\"xG\"\"#F(!\"\"" } }{PARA 11 "" 1 "" {XPPMATH 20 "6#-%$IntG6$,$*&\"\"#!\"\"%\"uG#\"\"\"F( F)F*" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,$*&\"\"$!\"\"%\"uG#F%\"\"#F& " }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,$*&\"\"$!\"\",&\"\"\"F(*$)%\"xG\" \"#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 10 "Exa mple 5 " }}{PARA 256 "" 0 "" {TEXT -1 24 " " } {XPPEDIT 18 0 "Int((2*x-1)/(x^2-x),x);" "6#-%$IntG6$*&,&*&\"\"#\"\"\"% \"xGF*F*F*!\"\"F*,&*$F+F)F*F+F,F,F+" }{TEXT -1 14 " --- " } {XPPEDIT 18 0 "PIECEWISE([u = x^2-x, ``],[``, ``],[du = (2*x-1)*dx, `` ]);" "6#-%*PIECEWISEG6%7$/%\"uG,&*$%\"xG\"\"#\"\"\"F+!\"\"%!G7$F/F/7$/ %#duG*&,&*&F,F-F+F-F-F-F.F-%#dxGF-F/" }{TEXT -1 2 " " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "`` = Int(1/u,u);" "6#/%!G-%$IntG6$*&\"\"\"F)%\"uG!\"\"F*" }{TEXT -1 1 " \+ " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 256 "" 0 "" {TEXT -1 1 " " } {XPPEDIT 18 0 "``=ln(abs(u))+c" "6#/%!G,&-%#lnG6#-%$absG6#%\"uG\"\"\"% \"cGF-" }{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 256 " " 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "``=ln(abs(x^2-x))+c" "6#/%!G,&-% #lnG6#-%$absG6#,&*$%\"xG\"\"#\"\"\"F.!\"\"F0%\"cGF0" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 119 "Int((2*x-1)/(x^2-x),x);\nsu bstitution := u=x^2-x;\nstudent[changevar](substitution,%%,u);\nvalue( %);\nsubs(substitution,%);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%$IntG6 $*&,&*&\"\"#\"\"\"%\"xGF*F*F*!\"\"F*,&*$)F+F)F*F*F+F,F,F+" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%-substitutionG/%\"uG,&*$)%\"xG\"\"#\"\"\"F,F* !\"\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%$IntG6$*&\"\"\"F'%\"uG!\"\" F(" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%#lnG6#%\"uG" }}{PARA 11 "" 1 " " {XPPMATH 20 "6#-%#lnG6#,&*$)%\"xG\"\"#\"\"\"F+F)!\"\"" }}}{PARA 0 " " 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 42 "Maple omits the ab solute value operation. " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 10 "Example 6 " }}{PARA 256 "" 0 "" {TEXT -1 24 " \+ " }{XPPEDIT 18 0 "Int(exp(-sin*x)*cos*x,x);" "6#-%$IntG6$* (-%$expG6#,$*&%$sinG\"\"\"%\"xGF-!\"\"F-%$cosGF-F.F-F." }{TEXT -1 14 " --- " }{XPPEDIT 18 0 "PIECEWISE([u = -sin*x, ``],[``, ``],[d u = -cos*x*dx, ``]);" "6#-%*PIECEWISEG6%7$/%\"uG,$*&%$sinG\"\"\"%\"xGF ,!\"\"%!G7$F/F/7$/%#duG,$*(%$cosGF,F-F,%#dxGF,F.F/" }{TEXT -1 2 " " } }{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 256 "" 0 "" {TEXT -1 1 " " } {XPPEDIT 18 0 "`` = -Int(exp(u),u);" "6#/%!G,$-%$IntG6$-%$expG6#%\"uGF ,!\"\"" }{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 256 " " 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "`` = -exp(u)+c;" "6#/%!G,&-%$exp G6#%\"uG!\"\"%\"cG\"\"\"" }{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "`` = -exp(-sin *x)+c;" "6#/%!G,&-%$expG6#,$*&%$sinG\"\"\"%\"xGF,!\"\"F.%\"cGF," } {TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT 259 4 "Note" }{TEXT -1 19 ": T he substitution " }{XPPEDIT 18 0 "u = sin*x;" "6#/%\"uG*&%$sinG\"\"\"% \"xGF'" }{TEXT -1 19 " may also be used. " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 124 "Int(exp(-sin(x))*cos (x),x);\nsubstitution := u=sin(x);\nstudent[changevar](substitution,%% ,u);\nvalue(%);\nsubs(substitution,%);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%$IntG6$*&-%$expG6#,$-%$sinG6#%\"xG!\"\"\"\"\"-%$cosGF-F0F." }} {PARA 11 "" 1 "" {XPPMATH 20 "6#>%-substitutionG/%\"uG-%$sinG6#%\"xG" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%$IntG6$-%$expG6#,$%\"uG!\"\"F*" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#,$-%$expG6#,$%\"uG!\"\"F)" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,$-%$expG6#,$-%$sinG6#%\"xG!\"\"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 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 13 "The integral " }{XPPEDIT 18 0 "Int(x*sqrt(2*x+3),x);" "6#-%$IntG6$*&%\"xG\"\"\"-%%sqrtG6#,&*&\" \"#F(F'F(F(\"\"$F(F(F'" }{TEXT -1 40 " can be found by using the subst itution " }{XPPEDIT 18 0 "u = 2*x+3;" "6#/%\"uG,&*&\"\"#\"\"\"%\"xGF(F (\"\"$F(" }{TEXT -1 2 ". " }}{PARA 256 "" 0 "" {TEXT -1 3 " " } {XPPEDIT 18 0 "Int(x*sqrt(2*x+3),x)" "6#-%$IntG6$*&%\"xG\"\"\"-%%sqrtG 6#,&*&\"\"#F(F'F(F(\"\"$F(F(F'" }{TEXT -1 11 " " }{XPPEDIT 18 0 "PIECEWISE([u = 2*x+3, x = (u-3)/2],[du = 2*dx, dx = ``(1/2)*`.`* du]);" "6#-%*PIECEWISEG6$7$/%\"uG,&*&\"\"#\"\"\"%\"xGF,F,\"\"$F,/F-*&, &F(F,F.!\"\"F,F+F27$/%#duG*&F+F,%#dxGF,/F7*(-%!G6#*&F,F,F+F2F,%\".GF,F 5F," }{TEXT -1 2 " " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 256 "" 0 "" {TEXT -1 4 "\004 = " }{XPPEDIT 18 0 "1/2" "6#*&\"\"\"F$\"\"#!\"\" " }{TEXT -1 1 " " }{XPPEDIT 18 0 "Int(``((u-3)/2)*sqrt(u),u);" "6#-%$I ntG6$*&-%!G6#*&,&%\"uG\"\"\"\"\"$!\"\"F-\"\"#F/F--%%sqrtG6#F,F-F," } {TEXT -1 1 " " }}{PARA 256 "" 0 "" {TEXT -1 3 " = " }{XPPEDIT 18 0 "1/ 2" "6#*&\"\"\"F$\"\"#!\"\"" }{TEXT -1 1 " " }{XPPEDIT 18 0 "Int(u*sqrt (u)/2-3*sqrt(u)/2,u)" "6#-%$IntG6$,&*(%\"uG\"\"\"-%%sqrtG6#F(F)\"\"#! \"\"F)*(\"\"$F)-F+6#F(F)F-F.F.F(" }{TEXT -1 1 " " }}{PARA 256 "" 0 "" {TEXT -1 3 " = " }{XPPEDIT 18 0 "1/2" "6#*&\"\"\"F$\"\"#!\"\"" }{TEXT -1 1 " " }{XPPEDIT 18 0 "Int(u^(3/2)/2-3*u^(1/2)/2,u)" "6#-%$IntG6$,&* &)%\"uG*&\"\"$\"\"\"\"\"#!\"\"F,F-F.F,*(F+F,)F)*&F,F,F-F.F,F-F.F.F)" } {TEXT -1 1 " " }}{PARA 256 "" 0 "" {TEXT -1 3 " = " }{XPPEDIT 18 0 "1/ 2" "6#*&\"\"\"F$\"\"#!\"\"" }{TEXT -1 1 " " }{XPPEDIT 18 0 "``(u^(5/2) /5-u^(3/2))+c;" "6#,&-%!G6#,&*&)%\"uG*&\"\"&\"\"\"\"\"#!\"\"F-F,F/F-)F **&\"\"$F-F.F/F/F-%\"cGF-" }{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 256 "" 0 "" {TEXT -1 3 " = " }{XPPEDIT 18 0 "u^(5/2)/10-u ^(3/2)/2 + c" "6#,(*&)%\"uG*&\"\"&\"\"\"\"\"#!\"\"F)\"#5F+F)*&)F&*&\" \"$F)F*F+F)F*F+F+%\"cGF)" }{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 256 "" 0 "" {TEXT -1 3 " = " }{XPPEDIT 18 0 "(2*x+3)^(5/2 )/10-(2*x+3)^(3/2)/2 + c" "6#,(*&),&*&\"\"#\"\"\"%\"xGF)F)\"\"$F)*&\" \"&F)F(!\"\"F)\"#5F.F)*&),&*&F(F)F*F)F)F+F)*&F+F)F(F.F)F(F.F.%\"cGF)" }{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 117 "Int(x*sqrt( 2*x+3),x);\nsubstitution := u=2*x+3;\nstudent[changevar](substitution, %%,u);\nvalue(%);\nsubs(substitution,%);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%$IntG6$*&%\"xG\"\"\",&*&\"\"#F(F'F(F(\"\"$F(#F(F+F'" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%-substitutionG/%\"uG,&*&\"\"#\"\"\"%\"xGF* F*\"\"$F*" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%$IntG6$,$*(\"\"#!\"\",& #\"\"$F(F)*&F(F)%\"uG\"\"\"F/F/F.#F/F(F/F." }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,&*&\"#5!\"\"%\"uG#\"\"&\"\"#\"\"\"*&F*F&F'#\"\"$F*F&" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,&*&\"#5!\"\",&*&\"\"#\"\"\"%\"xGF*F *\"\"$F*#\"\"&F)F**&F)F&F'#F,F)F&" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {PARA 0 "" 0 "" {TEXT -1 48 "We can get Maple to find this integral di rectly." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 31 "Int(x*sqrt(2*x+3),x);\nvalue(%);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%$IntG6$*&%\"xG\"\"\",&*&\"\"#F(F'F(F(\"\"$F(#F(F+F'" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,&*&\"#5!\"\",&*&\"\"#\"\"\"%\"xGF*F *\"\"$F*#\"\"&F)F**&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 10 "Example 8 " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 " ;" }}}{PARA 0 "" 0 "" {TEXT -1 13 "The integral " }{XPPEDIT 18 0 "Int( x/((1-3*x)^5),x)" "6#-%$IntG6$*&%\"xG\"\"\"*$,&F(F(*&\"\"$F(F'F(!\"\" \"\"&F-F'" }{TEXT -1 41 " can be found by using the substitution " } {XPPEDIT 18 0 "u = 1-3*x;" "6#/%\"uG,&\"\"\"F&*&\"\"$F&%\"xGF&!\"\"" } {TEXT -1 1 "." }}{PARA 256 "" 0 "" {TEXT -1 3 " " }{XPPEDIT 18 0 "In t(x/((1-3*x)^5),x);" "6#-%$IntG6$*&%\"xG\"\"\"*$,&F(F(*&\"\"$F(F'F(!\" \"\"\"&F-F'" }{TEXT -1 11 " " }{XPPEDIT 18 0 "PIECEWISE([u = 1-3*x, x = (1-u)/3],[du = -3*dx, dx = ``(-1/3)*`.`*du]);" "6#-%*PIECE WISEG6$7$/%\"uG,&\"\"\"F**&\"\"$F*%\"xGF*!\"\"/F-*&,&F*F*F(F.F*F,F.7$/ %#duG,$*&F,F*%#dxGF*F./F7*(-%!G6#,$*&F*F*F,F.F.F*%\".GF*F4F*" }{TEXT -1 2 " " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 256 "" 0 "" {TEXT -1 2 " " }{XPPEDIT 18 0 "`` = -1/3;" "6#/%!G,$*&\"\"\"F'\"\"$!\"\"F) " }{TEXT -1 1 " " }{XPPEDIT 18 0 "Int((1-u)/(3*u^5),u)" "6#-%$IntG6$*& ,&\"\"\"F(%\"uG!\"\"F(*&\"\"$F(*$F)\"\"&F(F*F)" }{TEXT -1 1 " " }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 256 "" 0 "" {TEXT -1 1 " " } {XPPEDIT 18 0 "`` = -1/9;" "6#/%!G,$*&\"\"\"F'\"\"*!\"\"F)" }{TEXT -1 1 " " }{XPPEDIT 18 0 "Int(u^(-5)-u^(-4),u)" "6#-%$IntG6$,&)%\"uG,$\"\" &!\"\"\"\"\")F(,$\"\"%F+F+F(" }{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 256 "" 0 "" {TEXT -1 2 " " }{XPPEDIT 18 0 "`` = -1/9; " "6#/%!G,$*&\"\"\"F'\"\"*!\"\"F)" }{TEXT -1 1 " " }{XPPEDIT 18 0 "``( u^(-4)/(-4)-u^(-3)/(-3))+c;" "6#,&-%!G6#,&*&)%\"uG,$\"\"%!\"\"\"\"\",$ F,F-F-F.*&)F*,$\"\"$F-F.,$F3F-F-F-F.%\"cGF." }{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 256 "" 0 "" {TEXT -1 3 " = " } {XPPEDIT 18 0 "u^(-4)/36-u^(-3)/27+c;" "6#,(*&)%\"uG,$\"\"%!\"\"\"\"\" \"#OF)F**&)F&,$\"\"$F)F*\"#FF)F)%\"cGF*" }{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 256 "" 0 "" {TEXT -1 2 " " }{XPPEDIT 18 0 "`` = 1/(36*(1-3*x)^4)-1/(27*(1-3*x)^3)+c;" "6#/%!G,(*&\"\"\"F'*&\"#OF '*$,&F'F'*&\"\"$F'%\"xGF'!\"\"\"\"%F'F/F'*&F'F'*&\"#FF'*$,&F'F'*&F-F'F .F'F/F-F'F/F/%\"cGF'" }{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 128 "Int(x/((1-3*x)^5),x);\nsu bstitution := u=1-3*x;\nstudent[changevar](substitution,%%,u);\nnormal (%);\nvalue(%);\nsubs(substitution,%);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%$IntG6$*&%\"xG\"\"\",&F(F(*&\"\"$F(F'F(!\"\"!\"&F'" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%-substitutionG/%\"uG,&\"\"\"F(*&\"\"$F(%\" xGF(!\"\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%$IntG6$,$*(\"\"$!\"\",& #\"\"\"F(F,*&F(F)%\"uGF,F)F,F.!\"&F)F." }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%$IntG6$,$*(\"\"*!\"\",&\"\"\"F)%\"uGF+F+F,!\"&F+F," }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,&*&\"\"\"F%*&\"#FF%)%\"uG\"\"$F%!\"\"F+*&F% F%*&\"#OF%)F)\"\"%F%F+F%" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,&*&\"\"\" F%*&\"#FF%),&F%F%*&\"\"$F%%\"xGF%!\"\"F+F%F-F-*&F%F%*&\"#OF%)F)\"\"%F% F-F%" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 39 " We can go straight to the last result. " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 32 "Int(x/((1-3*x)^5),x);\n \004value(%);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%$IntG6$*&%\"xG\"\" \"*$),&F(F(F'!\"$\"\"&F(!\"\"F'" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,&* &\"\"\"F%*$),&!\"\"F%%\"xG\"\"$\"\"%F%F)#F%\"#O*&F%F%*$)F(F+F%F)#F%\"# F" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}} }{SECT 1 {PARA 4 "" 0 "" {TEXT -1 14 "The integrals " }{XPPEDIT 18 0 " Int(sin^n*x,x);" "6#-%$IntG6$*&)%$sinG%\"nG\"\"\"%\"xGF*F+" }{TEXT -1 5 " and " }{XPPEDIT 18 0 "Int(cos^n*x,x);" "6#-%$IntG6$*&)%$cosG%\"nG \"\"\"%\"xGF*F+" }{TEXT -1 7 " where " }{TEXT 276 1 "n" }{TEXT -1 8 " \+ is odd " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 82 "We can integrate odd powers of sines and cosines by straightforwar d substitutions." }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}} {SECT 1 {PARA 4 "" 0 "" {TEXT -1 10 "Example 1 " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 13 "The integral " }{XPPEDIT 18 0 "Int(cos^5*x,x);" "6#-%$IntG6$*&%$cosG\"\"&%\"xG\"\"\"F)" }{TEXT -1 40 " can be found by using the substitution " }{XPPEDIT 18 0 "u = s in*x;" "6#/%\"uG*&%$sinG\"\"\"%\"xGF'" }{TEXT -1 3 ". " }}{PARA 256 " " 0 "" {TEXT -1 3 " " }{XPPEDIT 18 0 "Int(cos^5*x,x) = Int(cos^4*x*` .`*cos*x,x);" "6#/-%$IntG6$*&%$cosG\"\"&%\"xG\"\"\"F*-F%6$*,F(\"\"%F*F +%\".GF+F(F+F*F+F*" }{TEXT -1 11 " " }}{PARA 256 "" 0 "" {TEXT -1 2 " " }{XPPEDIT 18 0 "`` = Int((1-sin^2*x)^2*cos*x,x);" "6#/ %!G-%$IntG6$*(,&\"\"\"F**&%$sinG\"\"#%\"xGF*!\"\"F-%$cosGF*F.F*F." } {TEXT -1 13 " --- " }{XPPEDIT 18 0 "PIECEWISE([u = sin*x, ``], [``, ``],[du = cos*x*dx, ``]);" "6#-%*PIECEWISEG6%7$/%\"uG*&%$sinG\"\" \"%\"xGF+%!G7$F-F-7$/%#duG*(%$cosGF+F,F+%#dxGF+F-" }{TEXT -1 2 " " }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 256 "" 0 "" {TEXT -1 2 " " } {XPPEDIT 18 0 "`` = Int((1-u^2)^2,u);" "6#/%!G-%$IntG6$*$,&\"\"\"F**$% \"uG\"\"#!\"\"F-F," }{TEXT -1 1 " " }}{PARA 256 "" 0 "" {TEXT -1 2 " \+ " }{XPPEDIT 18 0 "`` = Int(``(1-2*u^2+u^4),u);" "6#/%!G-%$IntG6$-F$6#, (\"\"\"F+*&\"\"#F+*$%\"uGF-F+!\"\"*$F/\"\"%F+F/" }{TEXT -1 1 " " }} {PARA 256 "" 0 "" {TEXT -1 2 " " }{XPPEDIT 18 0 "`` = u-2*u^3/3+u^5/5 +c;" "6#/%!G,*%\"uG\"\"\"*(\"\"#F'*$F&\"\"$F'F+!\"\"F,*&F&\"\"&F.F,F'% \"cGF'" }{TEXT -1 2 " " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "`` = sin*x-2/3;" "6#/%!G,&*&%$sinG\"\"\"%\"xGF(F(*&\"\"#F(\"\"$! \"\"F-" }{TEXT -1 1 " " }{XPPEDIT 18 0 "sin^3*x+1/5;" "6#,&*&%$sinG\" \"$%\"xG\"\"\"F(*&F(F(\"\"&!\"\"F(" }{TEXT -1 1 " " }{XPPEDIT 18 0 "si n^5*x+c;" "6#,&*&%$sinG\"\"&%\"xG\"\"\"F(%\"cGF(" }{TEXT -1 2 ". " }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 121 "Int(cos(x)^5,x);\nsubstitution := u=sin(x);\nstudent[changevar](s ubstitution,%%,u);\nvalue(%);\nans1 := subs(substitution,%);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%$IntG6$*$)-%$cosG6#%\"xG\"\"&\"\"\"F+" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#>%-substitutionG/%\"uG-%$sinG6#%\"xG" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%$IntG6$*$),&\"\"\"F)*$)%\"uG\"\"#F )!\"\"F-F)F," }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,(%\"uG\"\"\"*&#F%\"\" &F%*$)F$F(F%F%F%*&#\"\"#\"\"$F%*$)F$F.F%F%!\"\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%%ans1G,(-%$sinG6#%\"xG\"\"\"*&#F*\"\"&F**$)F&F-F*F*F* *&#\"\"#\"\"$F**$)F&F3F*F*!\"\"" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {PARA 0 "" 0 "" {TEXT -1 38 "Maple's direct result looks different." } }{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 34 "Int(cos(x)^5,x);\nans2 := value(%);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%$IntG6$*$)-%$cosG6#%\"xG\"\"&\"\"\"F+" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%%ans2G,(*&#\"\"\"\"\"&F(*&)-%$cosG6#%\"xG\"\"%F( -%$sinGF.F(F(F(*&#F0\"#:F(*&)F,\"\"#F(F1F(F(F(*&#\"\")F5F(F1F(F(" }}} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 79 "We can di fferentiate and simplify to check that this second answer is correct. " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 44 "deriv := diff(ans2,x);\nsimplify(deriv,trig);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%&derivG,,*&#\"\"%\"\"&\"\"\"*&)-%$cosG6#%\"xG\" \"$F*)-%$sinGF/\"\"#F*F*!\"\"*&#F*F)F**$)F-F)F*F*F**&#\"\")\"#:F**&F-F *F2F*F*F6*&#F(F>F**$F,F*F*F**&#F=F>F*F-F*F*" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#*$)-%$cosG6#%\"xG\"\"&\"\"\"" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 102 "Another way to check that the \+ two answers are the same is to perform some trigonometric manipulation s." }}{PARA 0 "" 0 "" {TEXT -1 31 "For example, we can substitute " } {XPPEDIT 18 0 "cos*x = sqrt(1-sin^2*x);" "6#/*&%$cosG\"\"\"%\"xGF&-%%s qrtG6#,&F&F&*&%$sinG\"\"#F'F&!\"\"" }{TEXT -1 23 " in the second answe r. " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 46 "subs(cos(x)=sqrt(1-sin(x)^2),ans2);\nexpand(%);" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#,(*&#\"\"\"\"\"&F&*&),&F&F&*$)-%$sinG6 #%\"xG\"\"#F&!\"\"F1F&F-F&F&F&*&#\"\"%\"#:F&*&F*F&F-F&F&F&*&#\"\")F6F& F-F&F&" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,(-%$sinG6#%\"xG\"\"\"*&#F( \"\"&F(*$)F$F+F(F(F(*&#\"\"#\"\"$F(*$)F$F1F(F(!\"\"" }}}{PARA 0 "" 0 " " {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 30 "Alternatively, we can u se the " }{TEXT 0 7 "combine" }{TEXT -1 54 " procedure to express both answers in a different way." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 39 "combine(ans1,trig);\ncombine (ans2,trig);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,(*&#\"\"&\"\")\"\"\"- %$sinG6#%\"xGF(F(*&#F(\"#!)F(-F*6#,$*&F&F(F,F(F(F(F(*&#F&\"#[F(-F*6#,$ *&\"\"$F(F,F(F(F(F(" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,(*&#\"\"&\"\") \"\"\"-%$sinG6#%\"xGF(F(*&#F(\"#!)F(-F*6#,$*&F&F(F,F(F(F(F(*&#F&\"#[F( -F*6#,$*&\"\"$F(F,F(F(F(F(" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 10 "Example 2 " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 13 "The integral " }{XPPEDIT 18 0 "Int(sin^7*x,x);" "6#- %$IntG6$*&%$sinG\"\"(%\"xG\"\"\"F)" }{TEXT -1 40 " can be found by usi ng the substitution " }{XPPEDIT 18 0 "u = cos*x;" "6#/%\"uG*&%$cosG\" \"\"%\"xGF'" }{TEXT -1 1 "." }}{PARA 256 "" 0 "" {TEXT -1 3 " " } {XPPEDIT 18 0 "Int(sin^7*x,x) = Int(sin^6*x*`.`*sin*x,x);" "6#/-%$IntG 6$*&%$sinG\"\"(%\"xG\"\"\"F*-F%6$*,F(\"\"'F*F+%\".GF+F(F+F*F+F*" } {TEXT -1 11 " " }}{PARA 256 "" 0 "" {TEXT -1 5 " = " } {XPPEDIT 18 0 "Int((1-cos^2*x)^3*sin*x,x);" "6#-%$IntG6$*(,&\"\"\"F(*& %$cosG\"\"#%\"xGF(!\"\"\"\"$%$sinGF(F,F(F," }{TEXT -1 8 " " } {XPPEDIT 18 0 "PIECEWISE([u = cos*x, ``],[``, ``],[du = -sin*x*dx, ``] );" "6#-%*PIECEWISEG6%7$/%\"uG*&%$cosG\"\"\"%\"xGF+%!G7$F-F-7$/%#duG,$ *(%$sinGF+F,F+%#dxGF+!\"\"F-" }{TEXT -1 2 " " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "``=-Int((1-u^2)^3,u)" "6#/%!G,$-%$IntG6 $*$,&\"\"\"F+*$%\"uG\"\"#!\"\"\"\"$F-F/" }{TEXT -1 1 " " }}{PARA 256 " " 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "``=-Int(1-3*u^2+3*u^4-u^6,u)" "6 #/%!G,$-%$IntG6$,*\"\"\"F**&\"\"$F**$%\"uG\"\"#F*!\"\"*&F,F**$F.\"\"%F *F**$F.\"\"'F0F.F0" }{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "`` = Int(-1+3*u^2-3*u ^4+u^6,u);" "6#/%!G-%$IntG6$,*\"\"\"!\"\"*&\"\"$F)*$%\"uG\"\"#F)F)*&F, F)*$F.\"\"%F)F**$F.\"\"'F)F." }{TEXT -1 1 " " }}{PARA 256 "" 0 "" {TEXT -1 2 " " }{XPPEDIT 18 0 "`` = -u+u^3-3*u^5/5+u^7/7+c;" "6#/%!G, ,%\"uG!\"\"*$F&\"\"$\"\"\"*(F)F**$F&\"\"&F*F-F'F'*&F&\"\"(F/F'F*%\"cGF *" }{TEXT -1 1 " " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 " `` = -cos*x+cos^3*x-3/5;" "6#/%!G,(*&%$cosG\"\"\"%\"xGF(!\"\"*&F'\"\"$ F)F(F(*&F,F(\"\"&F*F*" }{TEXT -1 1 " " }{XPPEDIT 18 0 "cos^5*x+1/7;" " 6#,&*&%$cosG\"\"&%\"xG\"\"\"F(*&F(F(\"\"(!\"\"F(" }{TEXT -1 1 " " } {XPPEDIT 18 0 "cos^7*x+c;" "6#,&*&%$cosG\"\"(%\"xG\"\"\"F(%\"cGF(" } {TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 275 4 "Note" }{TEXT -1 14 ": The formula " }{XPPEDIT 18 0 "(1+v) ^3 = 1+3*v+3*v^2+v^3;" "6#/*$,&\"\"\"F&%\"vGF&\"\"$,*F&F&*&F(F&F'F&F&* &F(F&*$F'\"\"#F&F&*$F'F(F&" }{TEXT -1 23 " can be used to expand " } {XPPEDIT 18 0 "(1-u^2)^3" "6#*$,&\"\"\"F%*$%\"uG\"\"#!\"\"\"\"$" } {TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 121 "Int(sin(x)^7,x);\nsubstitution := u=cos(x);\nst udent[changevar](substitution,%%,u);\nvalue(%);\nans1 := subs(substitu tion,%);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%$IntG6$*$)-%$sinG6#%\"xG \"\"(\"\"\"F+" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%-substitutionG/%\"u G-%$cosG6#%\"xG" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%$IntG6$,$*$),&\" \"\"F**$)%\"uG\"\"#F*!\"\"\"\"$F*F/F-" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,*%\"uG!\"\"*&#\"\"\"\"\"(F(*$)F$F)F(F(F(*&#\"\"$\"\"&F(*$)F$F/F(F( F%*$)F$F.F(F(" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%%ans1G,*-%$cosG6#% \"xG!\"\"*&#\"\"\"\"\"(F-*$)F&F.F-F-F-*&#\"\"$\"\"&F-*$)F&F4F-F-F**$)F &F3F-F-" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 43 "We can check the answer by differentiation." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 70 "deriv := dif f(ans1,x);\nsubs(cos(x)=sqrt(1-sin(x)^2),deriv);\nexpand(%);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%&derivG,*-%$sinG6#%\"xG\"\"\"*&)-%$cosGF( \"\"'F*F&F*!\"\"*(\"\"$F*)F-\"\"%F*F&F*F**(F2F*)F-\"\"#F*F&F*F0" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#,*-%$sinG6#%\"xG\"\"\"*&),&F(F(*$)F$\" \"#F(!\"\"\"\"$F(F$F(F/*(F0F()F+F.F(F$F(F(*(F0F(F+F(F$F(F/" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#*$)-%$sinG6#%\"xG\"\"(\"\"\"" }}}{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 ";" }}}{PARA 0 "" 0 "" {TEXT -1 79 "In questi ons 1 to 16 find the given integral by using a suitable substitution. " }}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 3 "Q1 " }}{PARA 0 "" 0 "" {TEXT -1 2 " " }{XPPEDIT 18 0 "Int(x^3*(1-x^4)^6,x)" "6#-%$IntG6$*&%\"xG\" \"$,&\"\"\"F**$F'\"\"%!\"\"\"\"'F'" }{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{SECT 1 {PARA 5 "" 0 "" {TEXT -1 7 "Answer " }}{PARA 0 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "-1/28" "6#,$*&\"\"\"F%\"#G!\" \"F'" }{TEXT -1 1 " " }{XPPEDIT 18 0 "(1-x^4)^7+c" "6#,&*$,&\"\"\"F&*$ %\"xG\"\"%!\"\"\"\"(F&%\"cGF&" }{TEXT -1 1 " " }}}{PARA 0 "" 0 "" {TEXT -1 35 "___________________________________" }}{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 35 "____________ _______________________" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 "; " }}}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 3 "Q2 " }}{PARA 0 "" 0 "" {TEXT -1 2 " " }{XPPEDIT 18 0 "Int(x^2*sin(x^3),x)" "6#-%$IntG6$*&%\" xG\"\"#-%$sinG6#*$F'\"\"$\"\"\"F'" }{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{SECT 1 {PARA 5 "" 0 "" {TEXT -1 7 "Answer " }}{PARA 0 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "-cos(x^3)/3+c;" "6#,&*&-%$cos G6#*$%\"xG\"\"$\"\"\"F*!\"\"F,%\"cGF+" }{TEXT -1 1 " " }}}{PARA 0 "" 0 "" {TEXT -1 35 "___________________________________" }}{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 35 "__ _________________________________" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 3 "Q3 " }} {PARA 0 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "Int((4*x+1)*(2*x^2+x-5) ^3,x)" "6#-%$IntG6$*&,&*&\"\"%\"\"\"%\"xGF*F*F*F*F**$,(*&\"\"#F**$F+F/ F*F*F+F*\"\"&!\"\"\"\"$F*F+" }{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{SECT 1 {PARA 5 "" 0 "" {TEXT -1 7 "Answer " }}{PARA 0 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "1/4" "6#*&\"\"\"F$\"\"%!\"\"" } {TEXT -1 1 " " }{XPPEDIT 18 0 "(2*x^2+x-5)^4 +c" "6#,&*$,(*&\"\"#\"\" \"*$%\"xGF'F(F(F*F(\"\"&!\"\"\"\"%F(%\"cGF(" }{TEXT -1 1 " " }}}{PARA 0 "" 0 "" {TEXT -1 35 "___________________________________" }}{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 35 "__ _________________________________" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 3 "Q4 " }} {PARA 0 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "Int(x/sqrt(1+x^2),x)" " 6#-%$IntG6$*&%\"xG\"\"\"-%%sqrtG6#,&F(F(*$F'\"\"#F(!\"\"F'" }{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{SECT 1 {PARA 5 "" 0 "" {TEXT -1 7 "Answer " }}{PARA 0 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "sqrt(1 +x^2)+c;" "6#,&-%%sqrtG6#,&\"\"\"F(*$%\"xG\"\"#F(F(%\"cGF(" }{TEXT -1 1 " " }}}{PARA 0 "" 0 "" {TEXT -1 35 "________________________________ ___" }}{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 35 "___________________________________" }}{EXCHG {PARA 0 " > " 0 "" {MPLTEXT 1 0 1 ";" }}}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 3 "Q5 " }}{PARA 0 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "Int(cos*x/(3*sin*x +2),x);" "6#-%$IntG6$*(%$cosG\"\"\"%\"xGF(,&*(\"\"$F(%$sinGF(F)F(F(\" \"#F(!\"\"F)" }{TEXT -1 2 " " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {SECT 1 {PARA 5 "" 0 "" {TEXT -1 7 "Answer " }}{PARA 0 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "1/3" "6#*&\"\"\"F$\"\"$!\"\"" }{TEXT -1 1 " \+ " }{XPPEDIT 18 0 "ln(abs(3*sin*x+2))+c;" "6#,&-%#lnG6#-%$absG6#,&*(\" \"$\"\"\"%$sinGF-%\"xGF-F-\"\"#F-F-%\"cGF-" }{TEXT -1 1 " " }}}{PARA 0 "" 0 "" {TEXT -1 35 "___________________________________" }}{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 35 "__ _________________________________" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 3 "Q6 " }} {PARA 0 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "Int(x*exp(x^2),x)" "6#- %$IntG6$*&%\"xG\"\"\"-%$expG6#*$F'\"\"#F(F'" }{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{SECT 1 {PARA 5 "" 0 "" {TEXT -1 7 "Answer \+ " }}{PARA 0 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "1/2" "6#*&\"\"\"F$ \"\"#!\"\"" }{TEXT -1 1 " " }{XPPEDIT 18 0 "exp(x^2) +c" "6#,&-%$expG6 #*$%\"xG\"\"#\"\"\"%\"cGF*" }{TEXT -1 1 " " }}}{PARA 0 "" 0 "" {TEXT -1 35 "___________________________________" }}{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 35 "____________________ _______________" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}} {SECT 1 {PARA 4 "" 0 "" {TEXT -1 3 "Q7 " }}{PARA 0 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "Int(x*sqrt(1-2*x),x)" "6#-%$IntG6$*&%\"xG\"\"\"-%% sqrtG6#,&F(F(*&\"\"#F(F'F(!\"\"F(F'" }{TEXT -1 1 " " }}{PARA 0 "" 0 " " {TEXT -1 0 "" }}{SECT 1 {PARA 5 "" 0 "" {TEXT -1 7 "Answer " }} {PARA 0 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "1/10" "6#*&\"\"\"F$\"#5 !\"\"" }{TEXT -1 1 " " }{XPPEDIT 18 0 "(1-2*x)^(5/2)-1/6" "6#,&),&\"\" \"F&*&\"\"#F&%\"xGF&!\"\"*&\"\"&F&F(F*F&*&F&F&\"\"'F*F*" }{TEXT -1 1 " " }{XPPEDIT 18 0 "(1-2*x)^(3/2) + c" "6#,&),&\"\"\"F&*&\"\"#F&%\"xGF& !\"\"*&\"\"$F&F(F*F&%\"cGF&" }}}{PARA 0 "" 0 "" {TEXT -1 35 "_________ __________________________" }}{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 35 "___________________________________" }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{SECT 1 {PARA 4 "" 0 " " {TEXT -1 3 "Q8 " }}{PARA 0 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "In t(x/((4*x+3)^3),x)" "6#-%$IntG6$*&%\"xG\"\"\"*$,&*&\"\"%F(F'F(F(\"\"$F (F-!\"\"F'" }{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{SECT 1 {PARA 5 "" 0 "" {TEXT -1 7 "Answer " }}{PARA 0 "" 0 "" {TEXT -1 2 " \+ " }{XPPEDIT 18 0 "3/32/(4*x+3)^2-1/(16*(4*x+3))+c" "6#,(*(\"\"$\"\"\" \"#K!\"\"*$,&*&\"\"%F&%\"xGF&F&F%F&\"\"#F(F&*&F&F&*&\"#;F&,&*&F,F&F-F& F&F%F&F&F(F(%\"cGF&" }{TEXT -1 2 " " }}}{PARA 0 "" 0 "" {TEXT -1 35 " ___________________________________" }}{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 35 "_______________________________ ____" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 3 "Q9 " }}{PARA 0 "" 0 "" {TEXT -1 2 " " } {XPPEDIT 18 0 "Int((x+1)/sqrt(x^2+2*x+3),x)" "6#-%$IntG6$*&,&%\"xG\"\" \"F)F)F)-%%sqrtG6#,(*$F(\"\"#F)*&F/F)F(F)F)\"\"$F)!\"\"F(" }{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 35 "_________________________________ __" }}{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 35 "___________________________________" }}{EXCHG {PARA 0 " > " 0 "" {MPLTEXT 1 0 1 ";" }}}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 4 "Q1 0 " }}{PARA 0 "" 0 "" {TEXT -1 2 " " }{XPPEDIT 18 0 "Int(x^2/((x^3+2) ^(5/2)),x)" "6#-%$IntG6$*&%\"xG\"\"#),&*$F'\"\"$\"\"\"F(F-*&\"\"&F-F(! \"\"F0F'" }{TEXT -1 3 " " }}{PARA 0 "" 0 "" {TEXT -1 35 "___________ ________________________" }}{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 35 "___________________________________" }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{SECT 1 {PARA 4 "" 0 " " {TEXT -1 4 "Q11 " }}{PARA 0 "" 0 "" {TEXT -1 2 " " }{XPPEDIT 18 0 " Int((x+3)/sqrt(4*x-1),x)" "6#-%$IntG6$*&,&%\"xG\"\"\"\"\"$F)F)-%%sqrtG 6#,&*&\"\"%F)F(F)F)F)!\"\"F1F(" }{TEXT -1 3 " " }}{PARA 0 "" 0 "" {TEXT -1 35 "___________________________________" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 35 "____________ _______________________" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 "; " }}}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 4 "Q12 " }}{PARA 0 "" 0 "" {TEXT -1 2 " " }{XPPEDIT 18 0 "Int(cos^7*x,x);" "6#-%$IntG6$*&%$cosG \"\"(%\"xG\"\"\"F)" }{TEXT -1 3 " " }}{PARA 0 "" 0 "" {TEXT -1 35 "_ __________________________________" }}{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 35 "_______________________________ ____" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 4 "Q13 " }}{PARA 0 "" 0 "" {TEXT -1 2 " " } {XPPEDIT 18 0 "Int(cos^4*x*sin^3*x,x);" "6#-%$IntG6$**%$cosG\"\"%%\"xG \"\"\"%$sinG\"\"$F)F*F)" }{TEXT -1 3 " " }}{PARA 0 "" 0 "" {TEXT -1 35 "___________________________________" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 35 "____________________ _______________" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}} {SECT 1 {PARA 4 "" 0 "" {TEXT -1 4 "Q14 " }}{PARA 0 "" 0 "" {TEXT -1 2 " " }{XPPEDIT 18 0 "Int(sin(sqrt(x))/sqrt(x),x)" "6#-%$IntG6$*&-%$s inG6#-%%sqrtG6#%\"xG\"\"\"-F+6#F-!\"\"F-" }{TEXT -1 5 " " }}{PARA 0 "" 0 "" {TEXT -1 35 "___________________________________" }}{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 35 "__ _________________________________" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 4 "Q15 " }} {PARA 0 "" 0 "" {TEXT -1 3 " " }{XPPEDIT 18 0 "Int(sec^2*x*tan^2*x,x );" "6#-%$IntG6$**%$secG\"\"#%\"xG\"\"\"%$tanGF(F)F*F)" }{TEXT -1 13 " Hint: Let " }{XPPEDIT 18 0 "u = tan*x;" "6#/%\"uG*&%$tanG\"\"\"%\"x GF'" }{TEXT -1 4 " " }}{PARA 0 "" 0 "" {TEXT -1 35 "_______________ ____________________" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 " " {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" } }}{PARA 0 "" 0 "" {TEXT -1 35 "___________________________________" }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{SECT 1 {PARA 4 "" 0 " " {TEXT -1 4 "Q16 " }}{PARA 0 "" 0 "" {TEXT -1 4 " " }{XPPEDIT 18 0 "Int(sec^3*x*tan^3*x,x);" "6#-%$IntG6$**%$secG\"\"$%\"xG\"\"\"%$tanG F(F)F*F)" }{TEXT -1 13 " Hint: Let " }{XPPEDIT 18 0 "u = sec*x;" "6# /%\"uG*&%$secG\"\"\"%\"xGF'" }{TEXT -1 4 " " }}{PARA 0 "" 0 "" {TEXT -1 35 "___________________________________" }}{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 35 "____________ _______________________" }}{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 }