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}}}{SECT 1 {PARA 0 "" 0 "" {TEXT 257 2 "1." } {TEXT -1 106 " Compute the following indefinite integrals and check th e answers by differentiation and simplification.\n\n" }{TEXT 284 3 "(a )" }{TEXT -1 4 " " }{XPPEDIT 18 0 "int(sqrt(exp(x)-1),x)" "6#-%$int G6$-%%sqrtG6#,&-%$expG6#%\"xG\"\"\"F.!\"\"F-" }{TEXT -1 3 " \n\n" } {TEXT 285 3 "(b)" }{TEXT -1 3 " " }{XPPEDIT 18 0 "Int(x/(2*a*x-x^2)^ (3/2),x" "6#-%$IntG6$*&%\"xG\"\"\"),&*(\"\"#F(%\"aGF(F'F(F(*$F'F,!\"\" *&\"\"$F(F,F/F/F'" }{TEXT -1 3 " \n\n" }{TEXT 286 3 "(c)" }{TEXT -1 3 " " }{XPPEDIT 18 0 "Int(sqrt(x^2-a^2),x)" "6#-%$IntG6$-%%sqrtG6#,&*$ %\"xG\"\"#\"\"\"*$%\"aGF,!\"\"F+" }{TEXT -1 3 " \n\n" }{TEXT 287 3 "(d )" }{TEXT -1 3 " " }{XPPEDIT 18 0 "Int(1/x/sqrt(1+x^2),x" "6#-%$IntG 6$*(\"\"\"F'%\"xG!\"\"-%%sqrtG6#,&F'F'*$F(\"\"#F'F)F(" }{TEXT -1 2 "\n \n" }{TEXT 288 3 "(e)" }{TEXT -1 3 " " }{XPPEDIT 18 0 "Int(sec(x)^3, x)" "6#-%$IntG6$*$-%$secG6#%\"xG\"\"$F*" }{TEXT -1 2 "\n\n" }{TEXT 289 3 "(f)" }{TEXT -1 3 " " }{XPPEDIT 18 0 "Int(1/(1+sin(x)+cos(x)), x)" "6#-%$IntG6$*&\"\"\"F',(F'F'-%$sinG6#%\"xGF'-%$cosG6#F,F'!\"\"F," }{TEXT -1 2 "\n\n" }{TEXT 290 3 "(g)" }{TEXT -1 3 " " }{XPPEDIT 18 0 "Int(ln(x)/x/(x^2+1)^2,x)" "6#-%$IntG6$*(-%#lnG6#%\"xG\"\"\"F*!\"\"* $,&*$F*\"\"#F+F+F+F0F,F*" }{TEXT -1 1 "\n" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{SECT 1 {PARA 5 "" 0 "" {TEXT 291 3 "(a)" }}{EXCHG {PARA 0 "> \+ " 0 "" {MPLTEXT 1 0 8 "restart;" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 29 "Integrate(sqrt(exp(x)-1), x);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%$IntG6$*$,&-%$expG6#%\"xG\"\"\"F,!\"\"#F,\"\"#F+" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 9 "value(%);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,&*&\"\"#\"\"\",&-%$expG6#%\"xGF&F&!\"\"#F&F%F&*&F%F&-% 'arctanG6#*$F'F-F&F," }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 10 "dif f(%,x);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,&*&,&-%$expG6#%\"xG\"\"\"F *!\"\"#F+\"\"#F&F*F**&F*F**$F%#F*F-F+F+" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 10 "normal(%);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#*$,&-%$ expG6#%\"xG\"\"\"F)!\"\"#F)\"\"#" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}}{SECT 1 {PARA 5 "" 0 "" {TEXT 292 3 "(b)" }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 8 "restart;" }}}{EXCHG {PARA 0 " > " 0 "" {MPLTEXT 1 0 34 "Integrate(x/(2*a*x-x^2)^(3/2), x);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%$IntG6$*&%\"xG\"\"\",&*(\"\"#F(%\"aGF(F'F( F(*$)F'F+F(!\"\"#!\"$F+F'" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 9 "value(%);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,&*&\"\"\"F%*$,&*(\"\"#F %%\"aGF%%\"xGF%F%*$)F+F)F%!\"\"#F%F)F.F%**F)F.F*F.,&*&F)F%F*F%F%*&F)F% F+F%F.F%F'#F.F)F." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 10 "diff(% ,x);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,(*(\"\"#!\"\",&*(F%\"\"\"%\"a GF)%\"xGF)F)*$)F+F%F)F&#!\"$F%,&*&F%F)F*F)F)*&F%F)F+F)F&F)F&*&F)F)*&F* F)F'#F)F%F&F)**\"\"%F&F*F&F0F%F'F.F)" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 10 "normal(%);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#*&%\"xG \"\"\",&*(\"\"#F%%\"aGF%F$F%F%*$)F$F(F%!\"\"#!\"$F(" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}}{SECT 1 {PARA 5 "" 0 "" {TEXT 293 3 "(c)" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 8 "restart;" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 28 "Integrate(sqrt(x^2-a^2), x);" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#-%$IntG6$*$,&*$)%\"xG\"\"#\"\"\"F,*$)% \"aGF+F,!\"\"#F,F+F*" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 9 "valu e(%);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,&*(\"\"#!\"\"%\"xG\"\"\",&*$ )F'F%F(F(*$)%\"aGF%F(F&#F(F%F(*&#F(F%F(*&F-F(-%#lnG6#,&F'F(*$F)F/F(F(F (F&" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 10 "diff(%,x);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,(*&\"\"#!\"\",&*$)%\"xGF%\"\"\"F+*$)%\"aGF% F+F&#F+F%F+*(F%F&F*F%F'#F&F%F+**F%F&F.F%,&F+F+*&F'F1F*F+F+F+,&F*F+*$F' F/F+F&F&" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 13 "radnormal(%);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#*$,&*$)%\"xG\"\"#\"\"\"F)*$)%\"aGF(F )!\"\"#F)F(" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}}{SECT 1 {PARA 5 "" 0 "" {TEXT 294 4 "(d) " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 8 "restart;" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 33 "Integrate(1/(x*sqrt(1+x^2)), x );" }}{PARA 11 "" 1 "" {XPPMATH 20 "6# -%$IntG6$*&\"\"\"F'*&%\"xGF',&F'F'*$)F)\"\"#F'F'#F'F-!\"\"F)" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 9 "value(%);" }}{PARA 11 "" 1 " " {XPPMATH 20 "6#,$-%(arctanhG6#*&\"\"\"F(*$,&F(F(*$)%\"xG\"\"#F(F(#F( F.!\"\"F0" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 10 "diff(%,x);" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#*(,&\"\"\"F%*$)%\"xG\"\"#F%F%#!\"$F)F( F%,&F%F%*&F%F%F$!\"\"F.F." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 10 "normal(%);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#*&\"\"\"F$*&%\"xGF$, &F$F$*$)F&\"\"#F$F$#F$F*!\"\"" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}}{SECT 1 {PARA 5 "" 0 "" {TEXT 295 3 "(e)" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 8 "restart;" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 23 "Integrate(sec(x)^3, x);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%$IntG6$*$)-%$secG6#%\"xG\"\"$\"\"\"F+" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 9 "value(%);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,&*& #\"\"\"\"\"#F&*&-%$cosG6#%\"xG!\"#-%$sinGF+F&F&F&*&F%F&-%#lnG6#,&-%$se cGF+F&-%$tanGF+F&F&F&" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 10 "di ff(%,x);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,(*&-%$cosG6#%\"xG!\"$-%$s inGF'\"\"#\"\"\"*&#F-F,F-*&F-F-F%!\"\"F-F-*&F/F-*&,(*&-%$secGF'F--%$ta nGF'F-F-F-F-*$)F8F,F-F-F-,&F6F-F8F-F1F-F-" }}}{EXCHG {PARA 0 "> " 0 " " {MPLTEXT 1 0 12 "simplify(%);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#*& \"\"\"F$*$)-%$cosG6#%\"xG\"\"$F$!\"\"" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 23 "testeq( % = sec(x)^3 );" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%%trueG" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}} {SECT 1 {PARA 5 "" 0 "" {TEXT 296 3 "(f)" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 8 "restart;" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 34 "Integrate(1/(1+sin(x)+cos(x)), x);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6 #-%$IntG6$*&\"\"\"F',(F'F'-%$sinG6#%\"xGF'-%$cosGF+F'!\"\"F," }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 9 "value(%);" }}{PARA 11 "" 1 " " {XPPMATH 20 "6#-%#lnG6#,&-%$tanG6#,$*&\"\"#!\"\"%\"xG\"\"\"F/F/F/F/ " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 10 "diff(%,x);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#*&,&#\"\"\"\"\"#F&*&F%F&*$)-%$tanG6#,$*&F'!\"\" %\"xGF&F&F'F&F&F&F&,&F+F&F&F&F0" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 21 "convert(%, 'sincos');" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#*&,&# \"\"\"\"\"#F&*&F%F&*&,&F&F&-%$cosG6#%\"xG!\"\"F'-%$sinGF-!\"#F&F&F&,&* &F*F&F0F/F&F&F&F/" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 12 "simpli fy(%);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#*&\"\"\"F$,(F$F$-%$sinG6#%\" xGF$-%$cosGF(F$!\"\"" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }} }}{SECT 1 {PARA 5 "" 0 "" {TEXT 297 3 "(g)" }}{EXCHG {PARA 0 "> " 0 " " {MPLTEXT 1 0 8 "restart;" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 34 "Integrate(ln(x)/(x*(x^2+1)^2), x);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%$IntG6$*(-%#lnG6#%\"xG\"\"\"F*!\"\",&*$)F*\"\"#F+F+F+F+!\"#F* " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 9 "value(%);" }}{PARA 12 " " 1 "" {XPPMATH 20 "6#,0*&#\"\"\"\"\"#F&*&-%#lnG6#%\"xGF&-F*6#,&F&F&*& F,F&^#F&F&F&F&F&!\"\"*&#F&F'F&*&F)F&-F*6#,&F&F&*&^#F2F&F,F&F&F&F&F2*&# F&F'F&-%&dilogGF.F&F2*&#F&F'F&-F>F7F&F2*&#F&\"\"%F&-F*6#,&*$)F,F'F&F&F &F&F&F&*&#F&F'F&*(F)F&F,F'FGF2F&F2*&#F&F'F&*$)F)F'F&F&F&" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 10 "diff(%,x);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,,*(^##!\"\"\"\"#\"\"\"-%#lnG6#%\"xGF),&F)F)*&F-F)^#F)F )F)F'F)*(^##F)F(F)F*F),&F)F)*&^#F'F)F-F)F)F'F)*(F*F)F-F),&*$)F-F(F)F)F )F)F'F'*(F*F)F-\"\"$F8!\"#F)*&F*F)F-F'F)" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 22 "simplify(%, symbolic);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#*(-%#lnG6#%\"xG\"\"\"F'!\"\",&*$)F'\"\"#F(F(F(F(!\"#" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}}}{SECT 1 {PARA 0 "" 0 "" {TEXT 258 2 "2." }{TEXT -1 9 " Compute " }{XPPEDIT 18 0 "Int(x^n*exp(x ), x)" "6#-%$IntG6$*&)%\"xG%\"nG\"\"\"-%$expG6#F(F*F(" }{TEXT -1 21 " \+ for general integer " }{XPPEDIT 18 0 "n" "6#%\"nG" }{TEXT -1 44 " and \+ check the result for distinct value of " }{XPPEDIT 18 0 "n" "6#%\"nG" }{TEXT -1 2 ".\n" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 8 "restart; " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 24 "Integrate(x^n*exp(x),x) ;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%$IntG6$*&)%\"xG%\"nG\"\"\"-%$ex pG6#F(F*F(" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 29 "value(%) assu ming n::integer;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%*PIECEWISEG6%7$- %$expG6#%\"xG/%\"nG\"\"!7$*&-%$SumG6$**-%*factorialG6#F,\"\"\")!\"\",& F,F7%\"iGF9F7-F56#F;F9)F*F;F7/F;;F-F,F7F'F72F-F,7$,&*(-F56#,&F,F9F7F9F 9-F16$*&F>F9-F56#,&F;F7F7F9F7/F;;F7FGF7F'F7F9*&FEF9-%#EiG6$F7,$F*F9F7F 9%*otherwiseG" }}}{PARA 0 "" 0 "" {TEXT -1 36 "The antiderivative as a function of " }{TEXT 358 1 "n" }{TEXT -1 1 ":" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 222 "F := proc(n::integer)\n if n=0 then\n exp (x)\n elif n>0 then\n sum(n!*(-1)^(n-i)/i!*x^i, i=0..n)*exp(x)\n else\n -1/(-n-1)!*sum(1/(x^i)*(i-1)!, i=1..(-n-1))\n *exp(x ) - 1/(-n-1)!*Ei(1,-x)\n end if\nend proc:" }}}{PARA 0 "" 0 "" {TEXT -1 52 "Some results and comparison with distinct values of " } {XPPEDIT 18 0 "n" "6#%\"nG" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 19 "seq(F(n), n=-2..2);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6',&*&%\"xG! \"\"-%$expG6#F%\"\"\"F&-%#EiG6$F*,$F%F&F&,$F+F&F'*&,&F*F&F%F*F*F'F**&, (\"\"#F**&F4F*F%F*F&*$)F%F4F*F*F*F'F*" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 50 "\{seq(testeq(F(n)=int(x^n*exp(x), x)), n=-10..10)\}; " }}{PARA 11 "" 1 "" {XPPMATH 20 "6#<#%%trueG" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}}{SECT 1 {PARA 0 "" 0 "" {TEXT 260 2 "3." } {TEXT -1 1 " " }{TEXT 298 3 "(a)" }{TEXT -1 9 " Compute " }{XPPEDIT 18 0 "int(int((x-y)/(x+y)^3,y=0..1),x=0..1)" "6#-%$intG6$-F$6$*&,&%\"x G\"\"\"%\"yG!\"\"F+*$,&F*F+F,F+\"\"$F-/F,;\"\"!F+/F*;F3F+" }{TEXT -1 5 "\n\n " }{TEXT 299 3 "(b)" }{TEXT -1 9 " Compute " }{XPPEDIT 18 0 "int(int((x-y)/(x+y)^3,x=0..1),y=0..1)" "6#-%$intG6$-F$6$*&,&%\"xG\"\" \"%\"yG!\"\"F+*$,&F*F+F,F+\"\"$F-/F*;\"\"!F+/F,;F3F+" }{TEXT -1 5 "\n \n " }{TEXT 300 3 "(c)" }{TEXT -1 100 " Compare the results of (a) a nd (b). Does Maple make a mistake or is there something else going on? \n" }}{SECT 1 {PARA 5 "" 0 "" {TEXT 301 3 "(a)" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 8 "restart;" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 40 "Int(Int((x-y)/(x+y)^3, y=0..1), x=0..1);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%$IntG6$-F$6$*&,&%\"xG\"\"\"%\"yG!\"\"F+,&F*F+F,F+!\"$ /F,;\"\"!F+/F*F1" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 9 "value(%) ;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6##\"\"\"\"\"#" }}}{EXCHG {PARA 0 " > " 0 "" {MPLTEXT 1 0 0 "" }}}}{SECT 1 {PARA 5 "" 0 "" {TEXT 302 3 "(b )" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 8 "restart;" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 40 "Int(Int((x-y)/(x+y)^3, x=0..1), y=0 ..1);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%$IntG6$-F$6$*&,&%\"xG\"\"\" %\"yG!\"\"F+,&F*F+F,F+!\"$/F*;\"\"!F+/F,F1" }}}{EXCHG {PARA 0 "> " 0 " " {MPLTEXT 1 0 9 "value(%);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6##!\"\" \"\"#" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}}{SECT 1 {PARA 5 "" 0 "" {TEXT 353 3 "(c)" }}{PARA 0 "" 0 "" {TEXT -1 200 "Mapl e does not make a mistake. The conditions in Fubini's theorem, which s tates when integrals may be interchanged or not, are not satisfied. Th e integrand is not continuous on the area of integration" }}}}{SECT 1 {PARA 0 "" 0 "" {TEXT 259 2 "4." }{TEXT -1 44 " Compute the following \+ definite integrals.\n\n" }{TEXT 303 3 "(a)" }{TEXT -1 3 " " } {XPPEDIT 18 0 "Int((4*x^4+4*x^3-2*x^2-10*x+6)/(x^5+7*x^4+16*x^3+10*x^2 ),x=1..10)" "6#-%$IntG6$*&,,*&\"\"%\"\"\"*$%\"xGF)F*F**&F)F**$F,\"\"$F *F**&\"\"#F**$F,F1F*!\"\"*&\"#5F*F,F*F3\"\"'F*F*,**$F,\"\"&F**&\"\"(F* *$F,F)F*F**&\"#;F**$F,F/F*F**&F5F**$F,F1F*F*F3/F,;F*F5" }{TEXT -1 2 " \n\n" }{TEXT 304 3 "(b)" }{TEXT -1 3 " " }{XPPEDIT 18 0 "Int(x^4*sin (x)*cos(x),x=0..Pi/2)" "6#-%$IntG6$*(%\"xG\"\"%-%$sinG6#F'\"\"\"-%$cos G6#F'F,/F';\"\"!*&%#PiGF,\"\"#!\"\"" }{TEXT -1 2 "\n\n" }{TEXT 305 3 " (c)" }{TEXT -1 3 " " }{XPPEDIT 18 0 "Int(1/(x*sqrt(5*x^2-6*x+1)),x=1 /7..1/5)" "6#-%$IntG6$*&\"\"\"F'*&%\"xGF'-%%sqrtG6#,(*&\"\"&F'*$F)\"\" #F'F'*&\"\"'F'F)F'!\"\"F'F'F'F4/F);*&F'F'\"\"(F4*&F'F'F/F4" }{TEXT -1 2 "\n\n" }{TEXT 306 3 "(d)" }{TEXT -1 3 " " }{XPPEDIT 18 0 "Int(1/x, x=-2..-1)" "6#-%$IntG6$*&\"\"\"F'%\"xG!\"\"/F(;,$\"\"#F),$F'F)" } {TEXT -1 1 "\n" }}{SECT 1 {PARA 5 "" 0 "" {TEXT 307 3 "(a)" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 8 "restart;" }}}{EXCHG {PARA 0 "> " 0 " " {MPLTEXT 1 0 92 "Integrate((4*x^4 + 4*x^3 - 2*x^2 - 10*x + 6) / \n \+ (x^5 + 7*x^4 + 16*x^3 + 10*x^2), x=1..10);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%$IntG6$*&,,*&\"\"%\"\"\")%\"xGF)F*F**&F)F*)F,\"\"$F*F **&\"\"#F*)F,F1F*!\"\"*&\"#5F*F,F*F3\"\"'F*F*,**$)F,\"\"&F*F**&\"\"(F* F+F*F**&\"#;F*F.F*F**&F5F*F2F*F*F3/F,;F*F5" }}}{EXCHG {PARA 0 "> " 0 " " {MPLTEXT 1 0 9 "value(%);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,.#\"#F \"#]\"\"\"*&#\"$f\"F&F'-%#lnG6#\"\"#F'!\"\"*&#\"#>F&F'-F,6#\"\"&F'F/*& #\"#9F5F'-F,6#\"#6F'F'*&#\"$3$\"#DF'-%'arctanG6#\"#8F'F/*&#F>F?F'-FA6# \"\"%F'F'" }}}{PARA 0 "" 0 "" {TEXT -1 21 "Numerical validation:" }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 12 "evalf(%%=%);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/$\"+%*[+mB!\"*$\"**[+mB!\")" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}}{SECT 1 {PARA 5 "" 0 "" {TEXT 308 3 "(b) " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 8 "restart;" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 40 "Integrate(x^4*sin(x)*cos(x), x=0..P i/2);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%$IntG6$*()%\"xG\"\"%\"\"\"- %$sinG6#F(F*-%$cosGF-F*/F(;\"\"!,$*&\"\"#!\"\"%#PiGF*F*" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 11 "value(%); " }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,(*&#\"\"$\"#;\"\"\"*$)%#PiG\"\"#F(F(!\"\"*&#F(\"#kF(*$ )F+\"\"%F(F(F(#F&F3F(" }}}{PARA 0 "" 0 "" {TEXT -1 21 "Numerical valid ation:" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 12 "evalf(%%=%);" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#/$\"+AAm9U!#5$\"+?Am9UF&" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}}{SECT 1 {PARA 5 "" 0 "" {TEXT 309 3 "(c)" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 8 "restart;" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 47 "Integrate(1/(x*sqrt(5*x^2-6* x+1)), x=1/7..1/5);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%$IntG6$*&\"\" \"F'*&%\"xGF',(*&\"\"&F')F)\"\"#F'F'*&\"\"'F'F)F'!\"\"F'F'#F'F.F1/F);# F'\"\"(#F'F," }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 11 "value(%); \+ " }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%#ReG6#-%(arctanhG6#,$*(\"\"#\"\" \"\"\"$!\"\"F-#F,F+F," }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 9 "eva lc(%);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,$*&#\"\"\"\"\"%F&-%#lnG6#*& ,&*(\"\"#F&\"\"$!\"\"F/#F&F.F&F&F&F.,&*(F.F&F/F0F/F1F&F&F0!\"#F&F&" }} }{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 10 "expand(%);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,&*&#\"\"\"\"\"#F&-%#lnG6#,&*&F'F&\"\"$F%F&F-F&F&F &*&#F&F'F&-F)6#,&*&F'F&F-F%F&F-!\"\"F&F4" }}}{PARA 0 "" 0 "" {TEXT -1 21 "Numerical validation:" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 49 "factor(Int(1/(x*sqrt(5*x^2-6*x+1)), x=1/7..1/5));" }}{PARA 11 "" 1 " " {XPPMATH 20 "6#-%$IntG6$*&\"\"\"F'*&%\"xGF'*&,&*&\"\"&F'F)F'F'F'!\" \"F',&F)F'F'F.F'#F'\"\"#F./F);#F'\"\"(#F'F-" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 12 "evalf(%=%%);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/$ \"+(*y&pJ\"!\"*$\"+'*y&pJ\"F&" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}}{SECT 1 {PARA 5 "" 0 "" {TEXT 310 3 "(d)" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 8 "restart;" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 25 "Integrate(1/x, x=-2..-1);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%$IntG6$*&\"\"\"F'%\"xG!\"\"/F(;!\"#F)" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 9 "value(%);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,$-%#lnG6#\"\"#!\"\"" }}}{PARA 0 "" 0 "" {TEXT -1 21 "N umerical validation:" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 14 "eval f(%%=%); " }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/$!+1=ZJp!#5F$" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}}}{SECT 1 {PARA 0 "" 0 " " {TEXT 261 2 "5." }{TEXT -1 44 " Compute the following definite integ rals.\n\n" }{TEXT 311 3 "(a)" }{TEXT -1 3 " " }{XPPEDIT 18 0 "Int(1/ sqrt(1-x^2), x=0..1)" "6#-%$IntG6$*&\"\"\"F'-%%sqrtG6#,&F'F'*$%\"xG\" \"#!\"\"F//F-;\"\"!F'" }{TEXT -1 2 "\n\n" }{TEXT 312 3 "(b)" }{TEXT -1 3 " " }{XPPEDIT 18 0 "Int(x*arctan(x),x=0..1)" "6#-%$IntG6$*&%\"x G\"\"\"-%'arctanG6#F'F(/F';\"\"!F(" }{TEXT -1 2 "\n\n" }{TEXT 313 3 "( c)" }{TEXT -1 3 " " }{XPPEDIT 18 0 "Int(exp(-a*x)*cos(b*x)^2,x=0..in finity)" "6#-%$IntG6$*&-%$expG6#,$*&%\"aG\"\"\"%\"xGF-!\"\"F-*$-%$cosG 6#*&%\"bGF-F.F-\"\"#F-/F.;\"\"!%)infinityG" }{TEXT -1 24 ", for a posi tive number " }{XPPEDIT 18 0 "a" "6#%\"aG" }{TEXT -1 3 ".\n\n" }{TEXT 314 3 "(d)" }{TEXT -1 3 " " }{XPPEDIT 18 0 "Int(sin(x)/x,x=0..infini ty)" "6#-%$IntG6$*&-%$sinG6#%\"xG\"\"\"F*!\"\"/F*;\"\"!%)infinityG" } {TEXT -1 2 "\n\n" }{TEXT 315 3 "(e)" }{TEXT -1 3 " " }{XPPEDIT 18 0 "Int(exp(-x)*ln(x),x=0..infinity)" "6#-%$IntG6$*&-%$expG6#,$%\"xG!\"\" \"\"\"-%#lnG6#F+F-/F+;\"\"!%)infinityG" }{TEXT -1 2 "\n\n" }{TEXT 316 3 "(f)" }{TEXT -1 3 " " }{XPPEDIT 18 0 "Int(exp(-a*x)*ln(x)/sqrt(x), x=0..infinity)" "6#-%$IntG6$*(-%$expG6#,$*&%\"aG\"\"\"%\"xGF-!\"\"F-- %#lnG6#F.F--%%sqrtG6#F.F//F.;\"\"!%)infinityG" }{TEXT -1 29 ", for a p ositive real number " }{XPPEDIT 18 0 "a" "6#%\"aG" }{TEXT -1 3 ".\n\n " }{TEXT 317 3 "(g)" }{TEXT -1 3 " " }{XPPEDIT 18 0 "Int(exp(-sqrt(t ))/(t^(1/4)*(1-exp(-sqrt(t)))), t=0..infinity)" "6#-%$IntG6$*&-%$expG6 #,$-%%sqrtG6#%\"tG!\"\"\"\"\"*&)F.*&F0F0\"\"%F/F0,&F0F0-F(6#,$-F,6#F.F /F/F0F//F.;\"\"!%)infinityG" }{TEXT -1 2 "\n\n" }{TEXT 318 3 "(h)" } {TEXT -1 3 " " }{XPPEDIT 18 0 "Int(1/sqrt(x^4-1),x=1..infinity)" "6# -%$IntG6$*&\"\"\"F'-%%sqrtG6#,&*$%\"xG\"\"%F'F'!\"\"F//F-;F'%)infinity G" }{TEXT -1 2 "\n\n" }{TEXT 319 3 "(i)" }{TEXT -1 3 " " }{XPPEDIT 18 0 "Int(sqrt(cos(x)), x=0..Pi/2)" "6#-%$IntG6$-%%sqrtG6#-%$cosG6#%\" xG/F,;\"\"!*&%#PiG\"\"\"\"\"#!\"\"" }{TEXT -1 2 "\n\n" }{TEXT 320 3 "( j)" }{TEXT -1 3 " " }{XPPEDIT 18 0 "Int(1/sqrt(x^4+4*x^2+3), x=1..3) " "6#-%$IntG6$*&\"\"\"F'-%%sqrtG6#,(*$%\"xG\"\"%F'*&F.F'*$F-\"\"#F'F' \"\"$F'!\"\"/F-;F'F2" }{TEXT -1 2 "\n\n" }{TEXT 321 3 "(k)" }{TEXT -1 3 " " }{XPPEDIT 18 0 "Int(sqrt(tan(x)),x=0..Pi/2)" "6#-%$IntG6$-%%sq rtG6#-%$tanG6#%\"xG/F,;\"\"!*&%#PiG\"\"\"\"\"#!\"\"" }{TEXT -1 2 "\n\n " }{TEXT 322 3 "(l)" }{TEXT -1 3 " " }{XPPEDIT 18 0 "Int(sin(a*x)^2* sin(b*x)/x,x=0..infinity)" "6#-%$IntG6$*(-%$sinG6#*&%\"aG\"\"\"%\"xGF, \"\"#-F(6#*&%\"bGF,F-F,F,F-!\"\"/F-;\"\"!%)infinityG" }{TEXT -1 2 "\n \n" }{TEXT 323 3 "(m)" }{TEXT -1 3 " " }{XPPEDIT 18 0 "Int(1/cosh(a* x),x=0..infinity)" "6#-%$IntG6$*&\"\"\"F'-%%coshG6#*&%\"aGF'%\"xGF'!\" \"/F-;\"\"!%)infinityG" }{TEXT -1 29 ", for a positive real number " } {XPPEDIT 18 0 "a" "6#%\"aG" }{TEXT -1 2 ".\n" }}{SECT 1 {PARA 5 "" 0 " " {TEXT 324 3 "(a)" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 8 "restart ;" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 33 "Integrate(1/sqrt(1-x^2 ), x=0..1);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%$IntG6$*&\"\"\"F'*$,& F'F'*$)%\"xG\"\"#F'!\"\"#F'F-F./F,;\"\"!F'" }}}{EXCHG {PARA 0 "> " 0 " " {MPLTEXT 1 0 9 "value(%);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,$*&\" \"#!\"\"%#PiG\"\"\"F(" }}}{PARA 0 "" 0 "" {TEXT -1 21 "Numerical valid ation:" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 12 "evalf(%%=%);" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#/$\"+Fjzq:!\"*F$" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}}{SECT 1 {PARA 0 "" 0 "" {TEXT 256 3 "(b) " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 8 "restart;" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 31 "Integrate(x*arctan(x), x=0..1);" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#-%$IntG6$*&%\"xG\"\"\"-%'arctanG6#F'F( /F';\"\"!F(" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 9 "value(%);" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#,&*&\"\"%!\"\"%#PiG\"\"\"F(#F(\"\"#F& " }}}{PARA 0 "" 0 "" {TEXT -1 21 "Numerical validation:" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 12 "evalf(%%=%);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/$\"+M;)R&G!#5$\"+N;)R&GF&" }}}{EXCHG {PARA 0 "> " 0 " " {MPLTEXT 1 0 0 "" }}}}{SECT 1 {PARA 5 "" 0 "" {TEXT 325 3 "(c)" }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 8 "restart;" }}}{EXCHG {PARA 0 " > " 0 "" {MPLTEXT 1 0 48 "Integrate(exp(-a*x)*cos(b*x)^2, x=0..infinit y) ;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%$IntG6$*&-%$expG6#,$*&%\"aG \"\"\"%\"xGF-!\"\"F-)-%$cosG6#*&%\"bGF-F.F-\"\"#F-/F.;\"\"!%)infinityG " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 22 "value(%) assuming a>0; " }}{PARA 11 "" 1 "" {XPPMATH 20 "6#*(,&*$)%\"aG\"\"#\"\"\"F)*&F(F))% \"bGF(F)F)F),&F%F)*&\"\"%F)F+F)F)!\"\"F'F0" }}}{PARA 0 "" 0 "" {TEXT -1 21 "Numerical validation:" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 29 "evalf(subs(\{a=2,b=3\}, %%=%));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6 #/$\"++++]F!#5F$" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}} {SECT 1 {PARA 5 "" 0 "" {TEXT 326 3 "(d)" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 8 "restart;" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 35 "Integrate(sin(x)/x, x=0..infinity);" }}{PARA 11 "" 1 "" {XPPMATH 20 " 6#-%$IntG6$*&-%$sinG6#%\"xG\"\"\"F*!\"\"/F*;\"\"!%)infinityG" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 9 "value(%);" }}{PARA 11 "" 1 " " {XPPMATH 20 "6#,$*&\"\"#!\"\"%#PiG\"\"\"F(" }}}{PARA 0 "" 0 "" {TEXT -1 21 "Numerical validation:" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 12 "evalf(%%=%);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/$\"+ Fjzq:!\"*F$" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}}{SECT 1 {PARA 5 "" 0 "" {TEXT 327 3 "(e)" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 8 "restart;" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 40 "Integrate(exp(-x)*ln(x), x=0..infinity);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%$IntG6$*&-%$expG6#,$%\"xG!\"\"\"\"\"-%#lnG6#F+F-/F+; \"\"!%)infinityG" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 9 "value(%) ;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,$%&gammaG!\"\"" }}}{PARA 0 "" 0 "" {TEXT -1 21 "Numerical validation:" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 12 "evalf(%%=%);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/$!+ \\m:sd!#5F$" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}}{SECT 1 {PARA 5 "" 0 "" {TEXT 328 3 "(f)" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 8 "restart;" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 50 "Integrate(exp(-a*x)*ln(x)/sqrt(x), x=0..infinity);" }}{PARA 11 "" 1 " " {XPPMATH 20 "6#-%$IntG6$*(-%$expG6#,$*&%\"aG\"\"\"%\"xGF-!\"\"F--%#l nG6#F.F-F.#F/\"\"#/F.;\"\"!%)infinityG" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 22 "value(%) assuming a>0;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,(*(%\"aG#!\"\"\"\"#-%#lnG6#F%\"\"\"%#PiG#F,F(F'*(F%F&F-F.%&gamm aGF,F'**F(F,F%F&F-F.-F*6#F(F,F'" }}}{PARA 0 "" 0 "" {TEXT -1 21 "Numer ical validation:" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 22 "evalf(su bs(a=2,%%=%));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/$!+NgiHL!\"*F$" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}}{SECT 1 {PARA 5 "" 0 " " {TEXT 329 3 "(g)" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 8 "restart ;" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 70 "Integrate(exp(-sqrt(t) )/(t^(1/4)*(1-exp(-sqrt(t)))),\n t=0..infinity);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%$IntG6$*(-%$expG6#,$*$%\"tG#\"\"\"\"\"#!\"\"F.F,#F0\" \"%,&F.F.F'F0F0/F,;\"\"!%)infinityG" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 9 "value(%);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#*&%#PiG# \"\"\"\"\"#-%%ZetaG6##\"\"$F'F&" }}}{PARA 0 "" 0 "" {TEXT -1 21 "Numer ical validation:" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 12 "evalf(%% =%);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/$\"+ZZJIY!\"*$\"+[ZJIYF&" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}}{SECT 1 {PARA 5 "" 0 " " {TEXT 330 3 "(h)" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 8 "restart ;" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 40 "Integrate(1/sqrt(x^4-1 ), x=1..infinity);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%$IntG6$*&\"\" \"F'*$,&*$)%\"xG\"\"%F'F'F'!\"\"#F'\"\"#F./F,;F'%)infinityG" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 9 "value(%);" }}{PARA 11 "" 1 " " {XPPMATH 20 "6#,$*&#\"\"\"\"\"#F&*&F'F%-%*EllipticKG6#,$*&F'!\"\"F'F %F&F&F&F&" }}}{PARA 0 "" 0 "" {TEXT -1 21 "Numerical validation:" }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 12 "evalf(%%=%);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/$\"+x(G5J\"!\"*$\"+w(G5J\"F&" }}}{EXCHG {PARA 0 " > " 0 "" {MPLTEXT 1 0 0 "" }}}}{SECT 1 {PARA 5 "" 0 "" {TEXT 331 3 "(i )" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 8 "restart;" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 34 "Integrate(sqrt(cos(x)),x=0..Pi/2); " }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%$IntG6$*$-%$cosG6#%\"xG#\"\"\"\" \"#/F*;\"\"!,$*&F-!\"\"%#PiGF,F," }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 9 "value(%);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,&*&\"\"# #\"\"\"F%-%*EllipticKG6#,$*&F%!\"\"F%F&F'F'F-*(F%F'F%F&-%*EllipticEGF* F'F'" }}}{PARA 0 "" 0 "" {TEXT -1 21 "Numerical validation:" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 12 "evalf(%%=%);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/$\"+N-9)>\"!\"*F$" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}}{SECT 1 {PARA 5 "" 0 "" {TEXT 332 3 "(j)" }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 8 "restart;" }}}{EXCHG {PARA 0 " > " 0 "" {MPLTEXT 1 0 39 "Integrate(1/sqrt(x^4+4*x^2+3), x=1..3);" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#-%$IntG6$*&\"\"\"F'*$,(*$)%\"xG\"\"%F' F'*&F-F')F,\"\"#F'F'\"\"$F'#F'F0!\"\"/F,;F'F1" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 9 "value(%);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,&*& #\"\"\"\"\"$F&*&F'#F&\"\"#-%*EllipticFG6$,$*(F'F&\"#5!\"\"F0F)F&,$*&F' F1\"\"'F)F&F&F&F&*&#F&F'F&*&F'F)-F,6$,$*&F*F1F*F)F&F2F&F&F1" }}}{PARA 0 "" 0 "" {TEXT -1 21 "Numerical validation:" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 12 "evalf(%%=%);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/$ \"+)y%HLP!#5$\"+$z%HLPF&" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 " " }}}}{SECT 1 {PARA 5 "" 0 "" {TEXT 333 3 "(k)" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 8 "restart;" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 35 "Integrate(sqrt(tan(x)), x=0..Pi/2);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%$IntG6$*$-%$tanG6#%\"xG#\"\"\"\"\"#/F*;\"\"!,$*&F-!\" \"%#PiGF,F," }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 9 "value(%);" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#,$*(\"\"#!\"\"F%#\"\"\"F%%#PiGF(F(" }} }{PARA 0 "" 0 "" {TEXT -1 21 "Numerical validation:" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 12 "evalf(%%=%);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/$\"+p9W@A!\"*F$" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}}{SECT 1 {PARA 5 "" 0 "" {TEXT 334 3 "(l)" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 8 "restart;" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 48 "Integrate(sin(a*x)^2*sin(b*x)/x, x=0..infinity);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%$IntG6$*(-%$sinG6#*&%\"aG\"\"\"%\"xGF,\"\"#-F(6#*& %\"bGF,F-F,F,F-!\"\"/F-;\"\"!%)infinityG" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 9 "value(%);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,(*&#\"\" \"\"\")F&*&-%%csgnG6#,&%\"bGF&*&\"\"#F&%\"aGF&F&F&%#PiGF&F&!\"\"*&#F& \"\"%F&*&-F*6#F-F&F1F&F&F&*&#F&F'F&*&-F*6#,&F-F2*&F/F&F0F&F&F&F1F&F&F& " }}}{PARA 0 "" 0 "" {TEXT -1 21 "Numerical validation:" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 29 "evalf(subs(\{a=2,b=3\}, %%=%));" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#/$\"+M;)R&y!#5$\"+N;)R&yF&" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}}{SECT 1 {PARA 5 "" 0 "" {TEXT 335 3 "(m)" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 8 "restart;" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 38 "Integrate(1/cosh(a*x), x=0.. infinity);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%$IntG6$*&\"\"\"F'-%%co shG6#*&%\"aGF'%\"xGF'!\"\"/F-;\"\"!%)infinityG" }}}{EXCHG {PARA 0 "> \+ " 0 "" {MPLTEXT 1 0 22 "value(%) assuming a>0;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,$*(\"\"#!\"\"%#PiG\"\"\"%\"aGF&F(" }}}{PARA 0 "" 0 "" {TEXT -1 21 "Numerical validation:" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 23 "evalf(subs(a=2, %%=%));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/$\"+M;)R&y!#5$\"+N;)R&yF&" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}}}{SECT 1 {PARA 0 "" 0 "" {TEXT 262 2 "6." } {TEXT -1 5 " Let " }{XPPEDIT 18 0 "F" "6#%\"FG" }{TEXT -1 28 " be the \+ function defined by " }{XPPEDIT 18 0 "F(T):=Int(exp(-u^2*T)/u,u=1..T) " "6#>-%\"FG6#%\"TG-%$IntG6$*&-%$expG6#,$*&%\"uG\"\"#F'\"\"\"!\"\"F3F1 F4/F1;F3F'" }{TEXT -1 3 ".\n\n" }{TEXT 336 3 "(a)" }{TEXT -1 41 " Defi ne the corresponding Maple function " }{XPPEDIT 18 0 "F" "6#%\"FG" } {TEXT -1 42 " and determine a numeric approximation of " }{XPPEDIT 18 0 "F(2)" "6#-%\"FG6#\"\"#" }{TEXT -1 3 ".\n\n" }{TEXT 337 3 "(b)" } {TEXT -1 25 " Compute the derivative " }{XPPEDIT 18 0 "`F'`" "6#%#F'G " }{TEXT -1 5 " (by " }{TEXT 0 1 "D" }{TEXT -1 14 ") and compute " } {XPPEDIT 18 0 "`F'`(2)" "6#-%#F'G6#\"\"#" }{TEXT -1 2 ".\n" }}{SECT 1 {PARA 5 "" 0 "" {TEXT 338 3 "(a)" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 8 "restart;" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 44 "F := T - > evalf(Int(exp(-u^2*T)/u, u=1..T));" }}{PARA 11 "" 1 "" {XPPMATH 20 " 6#>%\"FGf*6#%\"TG6\"6$%)operatorG%&arrowGF(-%&evalfG6#-%$IntG6$*&-%$ex pG6#,$*&)%\"uG\"\"#\"\"\"9$F;!\"\"F;F9F=/F9;F;F " 0 "" {MPLTEXT 1 0 5 "F(2);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"+aA9VC!#6" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}} {SECT 1 {PARA 5 "" 0 "" {TEXT 339 3 "(b)" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 8 "restart;" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 44 "F := T -> evalf(Int(exp(-u^2*T)/u, u=1..T));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"FGf*6#%\"TG6\"6$%)operatorG%&arrowGF(-%&evalfG6#-%$ IntG6$*&-%$expG6#,$*&)%\"uG\"\"#\"\"\"9$F;!\"\"F;F9F=/F9;F;F " 0 "" {MPLTEXT 1 0 5 "D(F);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#f*6#%\"TG6\"6$%)operatorG%&arrowGF&-%&evalfG6#,&-%$IntG 6$,$*&%\"uG\"\"\"-%$expG6#,$*&)F3\"\"#F49$F4!\"\"F4F=/F3;F4F " 0 "" {MPLTEXT 1 0 8 "D(F)(2);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$!+%QA#eL!#6" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}}}{SECT 1 {PARA 0 "" 0 " " {TEXT 263 2 "7." }{TEXT -1 21 " Let A be the area \{ " }{XPPEDIT 18 0 "(x,y)*`in`*R^2" "6#*(6$%\"xG%\"yG\"\"\"%#inGF'%\"RG\"\"#" }{TEXT -1 3 " | " }{XPPEDIT 18 0 "1/2<=x*y" "6#1*&\"\"\"F%\"\"#!\"\"*&%\"xGF% %\"yGF%" }{XPPEDIT 18 0 "``<=2, 1<=x" "6$1%!G\"\"#1\"\"\"%\"xG" } {XPPEDIT 18 0 "``<=3" "6#1%!G\"\"$" }{TEXT -1 14 " \}. Compute\n " } {XPPEDIT 18 0 "Int(Int(exp(1/(x*y))/y^2/(x+1)^2,x),y=A..``" "6#-%$IntG 6$-F$6$*(-%$expG6#*&\"\"\"F-*&%\"xGF-%\"yGF-!\"\"F-*$F0\"\"#F1*$,&F/F- F-F-F3F1F//F0;%\"AG%!G" }{TEXT -1 2 ".\n" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 8 "restart;" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 67 "Int( Int( exp(1/(x*y))/(y^2*(x+1)^2), \n y=1/(2*x)..2/x), x=1..4 ); " }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%$IntG6$-F$6$*(-%$expG6#*&\"\"\"F -*&%\"xGF-%\"yGF-!\"\"F-F0!\"#,&F/F-F-F-F2/F0;,$*&F-F-*&\"\"#F-F/F-F1F -,$*&F9F-F/F1F-/F/;F-\"\"%" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 9 "value(%);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,.*&-%$expG6##\"\"\"\" \"#F)-%#lnG6#\"\"&F)!\"\"*&#\"\"$\"#5F)F%F)F)*&-F&6#F*F)F+F)F)*&#F2F3F )F5F)F/*&F%F)-F,F6F)F)*&F5F)F:F)F/" }}}{PARA 0 "" 0 "" {TEXT -1 21 "Nu merical validation:" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 12 "evalf (%%=%);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/$\"+_^rPN!\"*$\"+]^rPNF&" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}}{SECT 1 {PARA 0 "" 0 "" {TEXT 264 2 "8." }{TEXT -1 147 " In this exercise, we shall keep \+ track of the Risch algorithm and prove that the integral\n \+ " }{XPPEDIT 18 0 "Int((ln ^2*(x-1))/x,x)" "6#-%$IntG6$*(%#lnG\"\"#,&%\"xG\"\"\"F+!\"\"F+F*F,F*" }{TEXT -1 215 "\ndoes not exist in the class of elementary functions. \+ In the first place, Liouville's principle implies the following repres entation of the integral, when it would exist as an elemenatary functi on:\n " }{XPPEDIT 18 0 "B[3](x)*ln^3*(x-1)+B[2](x)*ln^ 2*(x-1)+B[1](x)*ln(x-1)+B[0](x)" "6#,**(-&%\"BG6#\"\"$6#%\"xG\"\"\"*$% #lnGF)F,,&F+F,F,!\"\"F,F,*(-&F'6#\"\"#6#F+F,*$F.F5F,,&F+F,F,F0F,F,*&-& F'6#F,6#F+F,-F.6#,&F+F,F,F0F,F,-&F'6#\"\"!6#F+F," }{TEXT -1 8 ",\nwher e " }{XPPEDIT 18 0 "B[3](x), B[2](x)" "6$-&%\"BG6#\"\"$6#%\"xG-&F%6#\" \"#6#F)" }{TEXT -1 6 ", and " }{XPPEDIT 18 0 "B[1](x)" "6#-&%\"BG6#\" \"\"6#%\"xG" }{TEXT -1 33 " are rational functions and only " } {XPPEDIT 18 0 "B[0](x)" "6#-&%\"BG6#\"\"!6#%\"xG" }{TEXT -1 42 " may c ontain new logarithmic extensions.\n\n" }{TEXT 340 3 "(a)" }{TEXT -1 51 " Determine the differential equations satisfied by " }{XPPEDIT 18 0 "B[0](x), `....`, B[3](x)" "6%-&%\"BG6#\"\"!6#%\"xG%%....G-&F%6#\"\" $6#F)" }{TEXT -1 3 ".\n\n" }{TEXT 341 3 "(b)" }{TEXT -1 11 " Show that " }{XPPEDIT 18 0 "B[3](x)" "6#-&%\"BG6#\"\"$6#%\"xG" }{TEXT -1 26 " i s a rational constant.\n\n" }{TEXT 342 3 "(c)" }{TEXT -1 42 " Prove th at the differential equation for " }{XPPEDIT 18 0 "B[2](x)" "6#-&%\"BG 6#\"\"#6#%\"xG" }{TEXT -1 67 " cannot be solved in terms of rational f unctions. This proves that " }{XPPEDIT 18 0 "(ln^2*(x-1))/x" "6#*(%#ln G\"\"#,&%\"xG\"\"\"F(!\"\"F(F'F)" }{TEXT -1 56 " cannot be integrated \+ in terms of elementary functions.\n" }}{SECT 1 {PARA 5 "" 0 "" {TEXT 354 3 "(a)" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 8 "restart;" }}} {PARA 0 "" 0 "" {TEXT -1 38 "We suppose the solution of given type:" } }{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 91 "Int( ln(x-1)^2/x, x ) = \nB [3](x)*ln(x-1)^3 + B[2](x)*ln(x-1)^2 +\nB[1](x)*ln(x-1) + B[0](x);" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#/-%$IntG6$*&-%#lnG6#,&%\"xG\"\"\"F-!\" \"\"\"#F,F.F,,**&-&%\"BG6#\"\"$6#F,F-)F(F6F-F-*&-&F46#F/F7F-)F(F/F-F-* &-&F46#F-F7F-F(F-F--&F46#\"\"!F7F-" }}}{PARA 0 "" 0 "" {TEXT -1 45 "Di fferentiation gives the following equation:" }}{EXCHG {PARA 0 "> " 0 " " {MPLTEXT 1 0 16 "map(diff, %, x);" }}{PARA 12 "" 1 "" {XPPMATH 20 "6 #/*&-%#lnG6#,&%\"xG\"\"\"F*!\"\"\"\"#F)F+,0*&-%%diffG6$-&%\"BG6#\"\"$6 #F)F)F*)F%F6F*F***F6F*F2F*F%F,F(F+F**&-F06$-&F46#F,F7F)F*)F%F,F*F***F, F*F=F*F%F*F(F+F**&-F06$-&F46#F*F7F)F*F%F*F**&FEF*F(F+F*-F06$-&F46#\"\" !F7F)F*" }}}{PARA 0 "" 0 "" {TEXT -1 80 "We bring all terms on one sid e of the equation and collect the logarithmic terms" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 18 "rhs(%)-lhs(%) = 0;" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#/,2*&-%%diffG6$-&%\"BG6#\"\"$6#%\"xGF/\"\"\")-%#lnG6#,& F/F0F0!\"\"F-F0F0**F-F0F)F0F2\"\"#F5F6F0*&-F'6$-&F+6#F8F.F/F0)F2F8F0F0 **F8F0F " 0 "" {MPLTEXT 1 0 20 "collect(%, ln(x-1));" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#/,,*&-%%di ffG6$-&%\"BG6#\"\"$6#%\"xGF/\"\"\")-%#lnG6#,&F/F0F0!\"\"F-F0F0*&,(*(F- F0F)F0F5F6F0-F'6$-&F+6#\"\"#F.F/F0*&F0F0F/F6F6F0)F2F?F0F0*&,&-F'6$-&F+ 6#F0F.F/F0*(F?F0F " 0 "" {MPLTEXT 1 0 19 "subs(ln(x-1)=u, %);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/,,*&-%%diffG6$-&%\"BG6#\"\"$6#%\"xGF/\"\"\")%\"uG F-F0F0*&,(*(F-F0F)F0,&F/F0F0!\"\"F7F0-F'6$-&F+6#\"\"#F.F/F0*&F0F0F/F7F 7F0)F2F=F0F0*&,&-F'6$-&F+6#F0F.F/F0*(F=F0F:F0F6F7F0F0F2F0F0*&FDF0F6F7F 0-F'6$-&F+6#\"\"!F.F/F0FN" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 50 "eqns := \{seq(coeftayl(lhs(%), u=0, k)=0, k=0..3)\};" }}{PARA 11 " " 1 "" {XPPMATH 20 "6#>%%eqnsG<&/-%%diffG6$-&%\"BG6#\"\"$6#%\"xGF0\"\" !/,(*(F.\"\"\"F*F5,&F0F5F5!\"\"F7F5-F(6$-&F,6#\"\"#F/F0F5*&F5F5F0F7F7F 1/,&-F(6$-&F,6#F5F/F0F5*(F=F5F:F5F6F7F5F1/,&*&FCF5F6F7F5-F(6$-&F,6#F1F /F0F5F1" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}}{SECT 1 {PARA 5 "" 0 "" {TEXT 355 3 "(b)" }}{PARA 0 "" 0 "" {TEXT -1 26 "From \+ part (a) we see that " }{XPPEDIT 18 0 "B[3](x)" "6#-&%\"BG6#\"\"$6#%\" xG" }{TEXT -1 41 " has to satisfy thedifferential equation " } {XPPEDIT 18 0 " diff(B[3](x),x)=0" "6#/-%%diffG6$-&%\"BG6#\"\"$6#%\"xG F-\"\"!" }{TEXT -1 29 ". The solution is a constant." }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 20 "diff(B[3](x),x) = 0;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%%diffG6$-&%\"BG6#\"\"$6#%\"xGF-\"\"!" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 19 "dsolve(%, B[3](x));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-&%\"BG6#\"\"$6#%\"xG%$_C1G" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}}{SECT 1 {PARA 5 "" 0 "" {TEXT 356 3 "(c) " }}{PARA 0 "" 0 "" {TEXT -1 4 "Let " }{XPPEDIT 18 0 "B[3](x) " "6#-&% \"BG6#\"\"$6#%\"xG" }{TEXT -1 26 " be the rational constant " } {XPPEDIT 18 0 "B[3]" "6#&%\"BG6#\"\"$" }{TEXT -1 80 ". We substitute t his in the differential equations that we obtained in part (a)." }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 31 "eval(subs(B[3](x)=B[3], eqns ));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#<&/,&-%%diffG6$-&%\"BG6#\"\"\"6 #%\"xGF/F-*(\"\"#F--&F+6#F1F.F-,&F/F-F-!\"\"F6F-\"\"!/,&*&F)F-F5F6F--F '6$-&F+6#F7F.F/F-F7/F7F7/,(*(\"\"$F-&F+6#FDF-F5F6F--F'6$F2F/F-*&F-F-F/ F6F6F7" }}}{PARA 0 "" 0 "" {TEXT -1 17 "So, the function " }{XPPEDIT 18 0 "B[2](x) " "6#-&%\"BG6#\"\"#6#%\"xG" }{TEXT -1 36 " satifies the \+ differential equation " }{XPPEDIT 18 0 "diff(B[2](x),x)+3*B[3]/(x-1)-1 /x = 0;" "6#/,(-%%diffG6$-&%\"BG6#\"\"#6#%\"xGF.\"\"\"*(\"\"$F/&F*6#F1 F/,&F.F/F/!\"\"F5F/*&F/F/F.F5F5\"\"!" }{TEXT -1 86 ".\nHowever, no rat ional function statisfies this equation; the solution is of the form \+ " }{XPPEDIT 18 0 "ln(x)+c" "6#,&-%#lnG6#%\"xG\"\"\"%\"cGF(" }{TEXT -1 1 "." }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 35 "diff(B[2](x),x)+3*B[ 3]/(x-1)-1/x=0;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/,(*(\"\"$\"\"\"&% \"BG6#F&F',&%\"xGF'F'!\"\"F-F'-%%diffG6$-&F)6#\"\"#6#F,F,F'*&F'F'F,F-F -\"\"!" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 19 "dsolve(%, B[2](x) );" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-&%\"BG6#\"\"#6#%\"xG,(*(\"\"$ \"\"\"&F&6#F-F.-%#lnG6#,&F*F.F.!\"\"F.F5-F2F)F.%$_C1GF." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}}}{SECT 1 {PARA 0 "" 0 "" {TEXT 265 2 "9." }{TEXT -1 44 " Show by the method of residues that\n \+ " }{XPPEDIT 18 0 "Int(1/(a^2*cos^2*theta +b^2*sin^2*theta), thet a=0..2*Pi)=2*Pi/(a*b)" "6#/-%$IntG6$*&\"\"\"F(,&*(%\"aG\"\"#%$cosGF,%& thetaGF(F(*(%\"bGF,%$sinGF,F.F(F(!\"\"/F.;\"\"!*&F,F(%#PiGF(*(F,F(F7F( *&F+F(F0F(F2" }{TEXT -1 7 " ,\nfor " }{XPPEDIT 18 0 "a, b" "6$%\"aG%\" bG" }{TEXT -1 23 " real and nonzero, and " }{XPPEDIT 18 0 "abs((b-a)/( b+a))<1" "6#2-%$absG6#*&,&%\"bG\"\"\"%\"aG!\"\"F*,&F)F*F+F*F,F*" } {TEXT -1 70 ". Compare your answer with Maple's answer when entering t he integral.\n" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 8 "restart;" } }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 25 "interface(showassumed=2): " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 71 "assume(a,real): additio nally(a<>0):\nassume(b,real): additionally(b<>0):" }}}{PARA 0 "" 0 "" {TEXT -1 55 "First we write our integrand as a rational funciton in " }{XPPEDIT 18 0 "z=exp(I*theta)" "6#/%\"zG-%$expG6#*&%\"IG\"\"\"%&theta GF*" }{TEXT -1 1 "." }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 24 "alias ( z=exp(I*theta) );" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%\"zG" }}} {PARA 0 "" 0 "" {TEXT -1 101 "Note the use of sqrt(-1). It is used bec ause you cannot define new aliases in terms of existing ones." }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 40 "1/(a^2*cos(theta)^2 + b^2*si n(theta)^2);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#*&\"\"\"F$,&*&)%#a|irG \"\"#F$)-%$cosG6#%&thetaGF)F$F$*&)%#b|irGF)F$)-%$sinGF-F)F$F$!\"\"" }} }{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 16 "convert(%, exp);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#*&\"\"\"F$,&*&)%#a|irG\"\"#F$),&*&F)!\"\"%\" zGF$F$*&F$F$*&F)F$F.F$F-F$F)F$F$*(\"\"%F-%#b|irGF),&F.F$*&F$F$F.F-F-F) F-F-" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 10 "factor(%);" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#,$**\"\"%\"\"\"%\"zG\"\"#,**&%#a|irGF& )F'F(F&F&F+F&%#b|irG!\"\"*&F,F&F-F&F&F.,*F*F&F+F&F-F&F/F.F.F&" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 19 "map(collect, %, z);" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#,$**\"\"%\"\"\"%\"zG\"\"#,(*&,&%#b|irG F&%#a|irGF&F&)F'F(F&F&F-F&F,!\"\"F/,(*&,&F-F&F,F/F&F.F&F&F,F&F-F&F/F& " }}}{PARA 0 "" 0 "" {TEXT -1 4 "From" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 15 "diff(z, theta);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#*& %\"zG\"\"\"^#F%F%" }}}{PARA 0 "" 0 "" {TEXT -1 126 "follows that our o riginal integration problem can be transformed into the contour integr al over the unit circle with integrand" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 18 "integrand := %%/%;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6 #>%*integrandG**^#!\"%\"\"\"%\"zGF(,(*&,&%#b|irGF(%#a|irGF(F()F)\"\"#F (F(F.F(F-!\"\"F1,(*&,&F.F(F-F1F(F/F(F(F-F(F.F(F1" }}}{PARA 0 "" 0 "" {TEXT -1 75 "Note that the factors in the denominator are interchanged when you replace " }{XPPEDIT 18 0 "b" "6#%\"bG" }{TEXT -1 4 " by " } {XPPEDIT 18 0 "-b" "6#,$%\"bG!\"\"" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 39 "The answer to the definite integral is " }{XPPEDIT 18 0 " 2*Pi*I*Sum(`residues inside the unit circle`" "6#**\"\"#\"\"\"%#PiGF%% \"IGF%-%$SumG6#%@residues~inside~the~unit~circleGF%" }{TEXT -1 111 ". \+ Therefore, we search for the roots of the denominator that lie in the \+ unit circle and compute their residues." }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 27 "solve(denom(integrand), z);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6&*&,&%#a|irG\"\"\"%#b|irG!\"\"F(,$*&F$F&,&F'F&F%F&F&F(#F &\"\"#,$F#F(*&F+F(F)F,,$F/F(" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 16 "map(x->x^2,[%]);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#7&,$*&,&%#a| irG\"\"\"%#b|irG!\"\"F*,&F)F(F'F(F(F*F$,$*&F+F*F&F(F*F," }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 15 "map(combine,%);" }}{PARA 11 "" 1 " " {XPPMATH 20 "6#7&,$*&,&%#a|irG\"\"\"%#b|irG!\"\"F*,&F)F(F'F(F(F*F$*& ,&F'F*F)F(F(F+F*F," }}}{PARA 0 "" 0 "" {TEXT -1 157 "Note that the 3rd expression is the reciprocal of the 1st one, and that the 4th express ion is the reciprocal of the 2nd one. Also note that when you replace \+ " }{XPPEDIT 18 0 "b" "6#%\"bG" }{TEXT -1 4 " by " }{XPPEDIT 18 0 "-b" "6#,$%\"bG!\"\"" }{TEXT -1 80 " the 1st and 3rd expression interchange , and the 2nd and 4th interchange as well" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 17 "r := map(sqrt,%);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6# >%\"rG7&*$,$*&,&%#a|irG\"\"\"%#b|irG!\"\"F-,&F,F+F*F+F+F-#F+\"\"#F&*$* &,&F*F-F,F+F+F.F-F/F1" }}}{PARA 0 "" 0 "" {TEXT -1 26 "Under the assum ption that " }{XPPEDIT 18 0 "0<=``" "6#1\"\"!%!G" }{XPPEDIT 18 0 "(b-a )/(a+b)<1" "6#2*&,&%\"bG\"\"\"%\"aG!\"\"F',&F(F'F&F'F)F'" }{TEXT -1 148 ", the first two roots are outside the unit circle, and the last t wo roots are inside the unit circle. We compute the residues of the la st two roots." }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 21 "subs(z=Z, i ntegrand);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#**^#!\"%\"\"\"%\"ZGF&,(* &,&%#b|irGF&%#a|irGF&F&)F'\"\"#F&F&F,F&F+!\"\"F/,(*&,&F,F&F+F/F&F-F&F& F+F&F,F&F/" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 19 "residue(%, Z= r[3]);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#*(^##!\"\"\"\"#\"\"\"%#a|irG F&%#b|irGF&" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 20 "residue(%%, \+ Z=r[4]);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#*(^##!\"\"\"\"#\"\"\"%#a|i rGF&%#b|irGF&" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 14 "2*Pi*I*(%+ %%);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,$**\"\"#\"\"\"%#PiGF&%#a|irG! \"\"%#b|irGF)F&" }}}{PARA 0 "" 0 "" {TEXT -1 24 "So, we have proved th at " }{XPPEDIT 18 0 " Int(1/(a^2*cos^2*theta +b^2*sin^2*theta), theta= 0..2*Pi)=2*Pi/(a*b) " "6#/-%$IntG6$*&\"\"\"F(,&*(%\"aG\"\"#%$cosGF,%&t hetaGF(F(*(%\"bGF,%$sinGF,F.F(F(!\"\"/F.;\"\"!*&F,F(%#PiGF(*(F,F(F7F(* &F+F(F0F(F2" }{TEXT -1 17 ", \nprovided that " }{XPPEDIT 18 0 "0<=``" "6#1\"\"!%!G" }{XPPEDIT 18 0 "(b-a)/(a+b)<1" "6#2*&,&%\"bG\"\"\"%\"aG! \"\"F',&F(F'F&F'F)F'" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 0 " " }}{PARA 0 "" 0 "" {TEXT -1 43 "The integration result under the cond ition " }{XPPEDIT 18 0 "0<=``" "6#1\"\"!%!G" }{XPPEDIT 18 0 "(a-b)/(a+ b)<1" "6#2*&,&%\"aG\"\"\"%\"bG!\"\"F',&F&F'F(F'F)F'" }{TEXT -1 73 " ca n be easily computed from the above intermediate results by replacing \+ " }{XPPEDIT 18 0 "b" "6#%\"bG" }{TEXT -1 4 " by " }{XPPEDIT 18 0 "-b" "6#,$%\"bG!\"\"" }{TEXT -1 63 " in the above calculation of residues f or the 3rd and 4th root." }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 26 " subs(z=Z,b=-b, integrand);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#**^#!\"% \"\"\"%\"ZGF&,(*&,&%#b|irGF&%#a|irGF&F&)F'\"\"#F&F&F,F&F+!\"\"F/,(*&,& F,F&F+F/F&F-F&F&F+F&F,F&F/" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 19 "residue(%, Z=r[1]);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#*(^##\"\"\" \"\"#F&%#a|irG!\"\"%#b|irGF)" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 20 "residue(%%, Z=r[2]);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#*(^##\" \"\"\"\"#F&%#a|irG!\"\"%#b|irGF)" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 27 "subs( b=-b, 2*Pi*I*(%+%%));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,$**\"\"#\"\"\"%#PiGF&%#a|irG!\"\"%#b|irGF)F&" }}} {PARA 0 "" 0 "" {TEXT -1 23 "So, we concluse that " }{XPPEDIT 18 0 " Int(1/(a^2*cos^2*theta +b^2*sin^2*theta), theta=0..2*Pi)=2*Pi/(a*b)" " 6#/-%$IntG6$*&\"\"\"F(,&*(%\"aG\"\"#%$cosGF,%&thetaGF(F(*(%\"bGF,%$sin GF,F.F(F(!\"\"/F.;\"\"!*&F,F(%#PiGF(*(F,F(F7F(*&F+F(F0F(F2" }{TEXT -1 7 " ,\nfor " }{XPPEDIT 18 0 "a, b" "6$%\"aG%\"bG" }{TEXT -1 23 " real \+ and nonzero, and " }{XPPEDIT 18 0 "abs((b-a)/(b+a))<1" "6#2-%$absG6#*& ,&%\"bG\"\"\"%\"aG!\"\"F*,&F)F*F+F*F,F*" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 70 "Let us compare it wi th the result of Maple when entering the integral." }}{EXCHG {PARA 0 " > " 0 "" {MPLTEXT 1 0 8 "restart;" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 67 "Integrate( 1/(a^2*cos(theta)^2 + b^2*sin(theta)^2),\n theta=0..2*Pi);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%$IntG6$*&\"\"\"F' ,&*&)%\"aG\"\"#F')-%$cosG6#%&thetaGF,F'F'*&)%\"bGF,F')-%$sinGF0F,F'F'! \"\"/F1;\"\"!,$*&F,F'%#PiGF'F'" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 9 "value(%);" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#*0%#PiG\"\"\",**(- %%csgnG6#*(%\"aGF%-%*conjugateG6#*$,(*$)F,\"\"#F%F%*&F4F%)%\"bGF4F%!\" \"*&F4F%,&*&F3F%F6F%F8*$)F7\"\"%F%F%#F%F4F8F?F%^#F%F%F%,(F2F%*&F4F%F6F %F8*&F4F%F:F?F%F?F6F%F%*(F(F%FAF?F:F?F%*(-F)6#*(F,F%-F.6#*$FAF?F%F@F%F %F1F?F:F?F8*(FFF%F1F?F6F%F%F%F,F8F1#F8F4F7!\"#FAFMF@F%" }}}{PARA 0 "" 0 "" {TEXT -1 49 "Only when assumptions are made on the parameters " } {XPPEDIT 18 0 "a" "6#%\"aG" }{TEXT -1 5 " and " }{XPPEDIT 18 0 "b" "6# %\"bG" }{TEXT -1 37 ", Maple can compute a simpler result." }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 25 "interface(showassumed=2):" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 35 "assume(a>0,b>0): additionall y(a>b):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 66 "Integrate(1/(a^2 *cos(theta)^2 + b^2*sin(theta)^2),\ntheta=0..2*Pi);" }}{PARA 11 "" 1 " " {XPPMATH 20 "6#-%$IntG6$*&\"\"\"F',&*&)%#a|irG\"\"#F')-%$cosG6#%&the taGF,F'F'*&)%#b|irGF,F')-%$sinGF0F,F'F'!\"\"/F1;\"\"!,$*&F,F'%#PiGF'F' " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 9 "value(%);" }}{PARA 11 " " 1 "" {XPPMATH 20 "6#,$**\"\"#\"\"\"%#a|irG!\"\"%#b|irGF(%#PiGF&F&" } }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}}{SECT 1 {PARA 0 "" 0 "" {TEXT 266 3 "10." }{TEXT -1 89 " Solve the following integral equat ion by Laplace transforms.\n " }{XPPEDIT 18 0 "sin(t)=Int(BesselJ(0,t-theta)*f(theta),theta=0..t)" "6#/-%$sinG6#% \"tG-%$IntG6$*&-%(BesselJG6$\"\"!,&F'\"\"\"%&thetaG!\"\"F1-%\"fG6#F2F1 /F2;F/F'" }{TEXT -1 1 "\n" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 8 " restart;" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 15 "with(inttrans); " }}{PARA 12 "" 1 "" {XPPMATH 20 "6#7/%)addtableG%(fourierG%+fourierco sG%+fouriersinG%'hankelG%(hilbertG%+invfourierG%+invhilbertG%+invlapla ceG%*invmellinG%(laplaceG%'mellinG%*savetableG" }}}{EXCHG {PARA 0 "> \+ " 0 "" {MPLTEXT 1 0 66 "eqn := sin(t) = Integrate(BesselJ(0,t-theta)*f (theta),theta=0..t);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%$eqnG/-%$sin G6#%\"tG-%$IntG6$*&-%(BesselJG6$\"\"!,&F)!\"\"%&thetaG\"\"\"F5-%\"fG6# F4F5/F4;F1F)" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 19 "laplace(eqn , t, s);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/*&\"\"\"F%,&*$)%\"sG\"\"# F%F%F%F%!\"\"*&-%(laplaceG6%-%\"fG6#%\"tGF3F)F%F&#F+F*" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 30 "isolate(%, laplace(f(t),t,s));" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#/-%(laplaceG6%-%\"fG6#%\"tGF*%\"sG*&\" \"\"F-*$,&*$)F+\"\"#F-F-F-F-#F-F2!\"\"" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 20 "invlaplace(%, s, t);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%\"fG6#%\"tG-%(BesselJG6$\"\"!F'" }}}{PARA 0 "" 0 "" {TEXT -1 47 "Let us verify the correctness of this solution." }}{EXCHG {PARA 0 "> \+ " 0 "" {MPLTEXT 1 0 24 "f := unapply(rhs(%), t);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"fGf*6#%\"tG6\"6$%)operatorG%&arrowGF(-%(BesselJG6$ \"\"!9$F(F(F(" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 4 "eqn;" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#/-%$sinG6#%\"tG-%$IntG6$*&-%(BesselJG6 $\"\"!,&F'!\"\"%&thetaG\"\"\"F3-F-6$F/F2F3/F2;F/F'" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 9 "value(%);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6 #/-%$sinG6#%\"tG-%&limitG6%,$*&,&F'!\"\"%&thetaG\"\"\"F0-%*hypergeomG6 %7$#F0\"\"#F57%F0F0#\"\"$F6,$*$)F-F6F0F.F0F./F/\"\"!%&rightG" }}} {PARA 0 "" 0 "" {TEXT -1 70 "If no symbolic verifiaction is possible, \+ we do a graphical validation:" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 53 "plot( rhs(%%)-lhs(%%), t=0..2*Pi, -1..1, axes=frame);" }}{PARA 13 "" 1 "" {GLPLOT2D 293 293 293 {PLOTDATA 2 "6'-%'CURVESG6#7gn7$$\"\" !F)F(7$$\"+&eb&p8!#5F(7$$\"+w&)>hDF-F(7$$\"+]dK,RF-F(7$$\"+dhL]_F-F(7$ $\"+`\\$Hf'F-F(7$$\"+,fpPyF-F(7$$\"+jPdE\"*F-F(7$$\"+4L&f/\"!\"*F(7$$ \"+g<#)y6FCF(7$$\"+%H$\\:8FCF(7$$\"+bO(eV\"FCF(7$$\"+IMRr:FCF(7$$\"+i' puq\"FCF(7$$\"+EVgQ=FCF(7$$\"+jrod>FCF(7$$\"+v\")G*4#FCF(7$$\"+1GC>AFC F(7$$\"+f&y(eBFCF(7$$\"+V8H#[#FCF(7$$\"+VW!yh#FCF(7$$\"+h\\%ou#FCF(7$$ \"+._[\")GFCF(7$$\"+wp70IFCF(7$$\"+K8\\QJFCF(7$$\"+;%>qF$FCF(7$$\"+?&3 wR$FCF(7$$\"+c![y_$FCF(7$$\"+T#)RiOFCF(7$$\"+Z$HSz$FCF(7$$\"+6#*Q@RFCF (7$$\"+D1!G1%FCF(7$$\"+q]')*=%FCF(7$$\"+([LbK%FCF(7$$\"+O#p%[WFCF(7$$ \"+#[qGe%FCF(7$$\"+\"eJ$4ZFCF(7$$\"+n)>:%[FCF(7$$\"+y!e2(\\FCF(7$$\"+3 &eg5&FCF(7$$\"+.*ojB&FCF(7$$\"+.,jp`FCF(7$$\"+Lyy,bFCF(7$$\"+hqABcFCF( 7$$\"+A,TidFCF(7$$\"+5r*o)eFCF(7$$\"+/ii>gFCF(7$$\"+Wb9$3'FCF(7$$\"+$) [mYhFCF(7$$\"+()\\z!='FCF(7$$\"+!4D\\@'FCF(7$$\"+T,*>B'FCF(7$$\"+$>b! \\iFCF(7$$\"+=xediFCF(7$$\"+W-7miFCF(7$$\"+2lQqiFCF(7$$\"+qFluiFCF(7$$ \"+L!>*yiFCF(7$$\"+&H&=$G'FC$!)]N+Q!#<-%*AXESSTYLEG6#%&FRAMEG-%+AXESLA BELSG6$Q\"t6\"Q!F_w-%'COLOURG6&%$RGBGF)F)F)-%%VIEWG6$;F($\"+3`=$G'FC;$ !\"\"F)$\"\"\"F)" 1 2 0 1 10 0 2 6 1 3 2 1.000000 45.000000 45.000000 0 0 "Curve 1" }}}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}} {SECT 1 {PARA 0 "" 0 "" {TEXT 267 3 "11." }{TEXT -1 29 " Solve the int egral equation " }{XPPEDIT 18 0 "f(t)=1+int((t-theta)*f(theta),theta=0 ..t)" "6#/-%\"fG6#%\"tG,&\"\"\"F)-%$intG6$*&,&F'F)%&thetaG!\"\"F)-F%6# F/F)/F/;\"\"!F'F)" }{TEXT -1 2 ".\n" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 8 "restart;" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 15 "with(inttrans);" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#7/%)addtableG%(fou rierG%+fouriercosG%+fouriersinG%'hankelG%(hilbertG%+invfourierG%+invhi lbertG%+invlaplaceG%*invmellinG%(laplaceG%'mellinG%*savetableG" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 57 "eqn := f(t)=1 + Integrate((t -theta)*f(theta),theta=0..t);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%$eq nG/-%\"fG6#%\"tG,&\"\"\"F+-%$IntG6$*&,&F)F+%&thetaG!\"\"F+-F'6#F1F+/F1 ;\"\"!F)F+" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 19 "laplace(eqn, \+ t, s);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%(laplaceG6%-%\"fG6#%\"tGF *%\"sG,&*&\"\"\"F.F+!\"\"F.*&F$F.F+!\"#F." }}}{EXCHG {PARA 0 "> " 0 " " {MPLTEXT 1 0 30 "isolate(%, laplace(f(t),t,s));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%(laplaceG6%-%\"fG6#%\"tGF*%\"sG*&\"\"\"F-*&F+F-,&F-F -*&F-F-*$)F+\"\"#F-!\"\"F4F-F4" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 10 "normal(%);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%(laplaceG6%-% \"fG6#%\"tGF*%\"sG*&F+\"\"\",&*$)F+\"\"#F-F-F-!\"\"F2" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 20 "invlaplace(%, s, t);" }}{PARA 11 " " 1 "" {XPPMATH 20 "6#/-%\"fG6#%\"tG-%%coshGF&" }}}{PARA 0 "" 0 "" {TEXT -1 47 "Let us verify the correctness of this solution." }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 24 "f := unapply(rhs(%), t);" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"fG%%coshG" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 4 "eqn;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%%coshG 6#%\"tG,&\"\"\"F)-%$IntG6$*&,&F'F)%&thetaG!\"\"F)-F%6#F/F)/F/;\"\"!F'F )" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 9 "value(%);" }}{PARA 11 " " 1 "" {XPPMATH 20 "6#/-%%coshG6#%\"tGF$" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}}{SECT 1 {PARA 0 "" 0 "" {TEXT 268 3 "12." } {TEXT -1 9 " Compute " }{XPPEDIT 18 0 "Int(x/(1+x^2)*sin(omega^2*x),x= 0..infinity)" "6#-%$IntG6$*(%\"xG\"\"\",&F(F(*$F'\"\"#F(!\"\"-%$sinG6# *&%&omegaGF+F'F(F(/F';\"\"!%)infinityG" }{TEXT -1 1 "\n" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 8 "restart;" }}}{EXCHG {PARA 0 "> " 0 " " {MPLTEXT 1 0 50 "Integrate(x/(1+x^2)*sin(omega^2*x),x=0..infinity); " }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%$IntG6$*(%\"xG\"\"\",&F(F(*$)F' \"\"#F(F(!\"\"-%$sinG6#*&)%&omegaGF,F(F'F(F(/F';\"\"!%)infinityG" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 9 "value(%);" }}{PARA 11 "" 1 " " {XPPMATH 20 "6#,(*&#\"\"\"\"\"#F&*(-%%csgnG6#*$)%&omegaGF'F&F&%#PiGF &-%%coshGF+F&F&F&*(^##!\"\"F'F&-%#CiG6#*&^#F5F&F-F&F&-%%sinhGF+F&F&*(^ #F%F&-F76#*&F-F&^#F&F&F&F;F&F&" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 9 "evalc(%);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,(*(^##!\"\"\"\"# \"\"\"-%#CiG6#*&^#F'F))%&omegaGF(F)F)-%%sinhG6#*$F/F)F)F)*(^##F)F(F)-F +6#*&F/F)^#F)F)F)F1F)F)*&F7F)*&%#PiGF)-%%coshGF3F)F)F)" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 17 "collect(%, sinh);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,&*&,&*&^##!\"\"\"\"#\"\"\"-%#CiG6#*&^#F)F+)%&omegaG F*F+F+F+*&^##F+F*F+-F-6#*&F1F+^#F+F+F+F+F+-%%sinhG6#*$F1F+F+F+*&F5F+*& %#PiGF+-%%coshGF " 0 "" {MPLTEXT 1 0 31 "expr := \+ coeff(%,sinh(omega^2));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%%exprG,&* &^##!\"\"\"\"#\"\"\"-%#CiG6#*&^#F)F+)%&omegaGF*F+F+F+*&^##F+F*F+-F-6#* &F1F+^#F+F+F+F+" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 30 "simplify (eval(expr, omega=1));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,$*&\"\"#!\" \"%#PiG\"\"\"F&" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 31 "simplify (eval(expr, omega=-1));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,$*&\"\"#! \"\"%#PiG\"\"\"F&" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 32 "simpli fy(eval(expr, omega=2/3));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,$*&\"\" #!\"\"%#PiG\"\"\"F&" }}}{PARA 0 "" 0 "" {TEXT -1 32 "You may guess tha t the value is " }{XPPEDIT 18 0 "-Pi/2" "6#,$*&%#PiG\"\"\"\"\"#!\"\"F( " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 18 "diff(expr, omega);" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#\"\"!" }}}{PARA 0 "" 0 "" {TEXT -1 31 "So, the expression is constant." }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 26 "eval(subs(omega=1, expr));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6# ,&*&^##!\"\"\"\"#\"\"\",&-%#CiG6#^#F)F)*&^#F'F)%#PiGF)F)F)F)*&^##F)F(F )F+F)F)" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 10 "expand(%);" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#,$*&\"\"#!\"\"%#PiG\"\"\"F&" }}}{PARA 0 "" 0 "" {TEXT -1 20 "The constant equals " }{XPPEDIT 18 0 "-Pi/2" "6 #,$*&%#PiG\"\"\"\"\"#!\"\"F(" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 42 "Alternatively, you could do the following:" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 5 "expr;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,&*&^ ##!\"\"\"\"#\"\"\"-%#CiG6#*&^#F'F))%&omegaGF(F)F)F)*&^##F)F(F)-F+6#*&F /F)^#F)F)F)F)" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 15 "convert(%, Ei);" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#,&*&^##!\"\"\"\"#\"\"\",**&#F )F(F)-%#EiG6$F)*$)%&omegaGF(F)F)F'*&#F)F(F)-F.6$F),$F0F'F)F'**F%F)-%%c sgnG6#F0F)%#PiGF)-F:6#*&F1F)^#F)F)F)F)*(F%F)F9F)F " 0 "" {MPLTEXT 1 0 9 "evalc(%);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,$*&\"\"#!\"\"%#PiG\"\"\"F&" }}}{EXCHG {PARA 0 "> " 0 " " {MPLTEXT 1 0 0 "" }}}}{SECT 1 {PARA 0 "" 0 "" {TEXT 269 3 "13." } {TEXT -1 9 " Compute " }{XPPEDIT 18 0 "int(1/((1+k^2*sin(t)^3)^3*sqrt( 1+k^2*sin(t)^2)),t=0..Pi/4)" "6#-%$intG6$*&\"\"\"F'*&,&F'F'*&%\"kG\"\" #-%$sinG6#%\"tG\"\"$F'F1-%%sqrtG6#,&F'F'*&F+F,-F.6#F0F,F'F'!\"\"/F0;\" \"!*&%#PiGF'\"\"%F9" }{TEXT -1 1 "\n" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 8 "restart;" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 19 "infolevel[int]:= 2:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 70 "Int egrate(1/((1+k^2*sin(t)^3)^3 * sqrt(1+k^2*sin(t)^2)),\n t=0..Pi/4);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%$IntG6$*&\"\"\"F'*&),&F'F'*&)%\"kG \"\"#F')-%$sinG6#%\"tG\"\"$F'F'F4F',&F'F'*&F,F')F0F.F'F'#F'F.!\"\"/F3; \"\"!,$*&\"\"%F9%#PiGF'F'" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 9 "value(%);" }}{PARA 6 "" 1 "" {TEXT -1 30 "int/ellalg/trxstandard/sign um:" }}{PARA 6 "" 1 "" {TEXT -1 6 "NoName" }}{PARA 6 "" 1 "" {TEXT -1 44 "Tried to determine the sign of --> 2*Im(1/k)" }}{PARA 6 "" 1 "" {TEXT -1 48 "int/indef1: first-stage indefinite integration" }} {PARA 6 "" 1 "" {TEXT -1 49 "int/indef2: second-stage indefinite int egration" }}{PARA 6 "" 1 "" {TEXT -1 48 "int/trigon: case of integra nd containing trigs" }}{PARA 6 "" 1 "" {TEXT -1 48 "int/indef1: firs t-stage indefinite integration" }}{PARA 6 "" 1 "" {TEXT -1 49 "int/alg ebraic2/algebraic: algebraic integration" }}{PARA 7 "" 1 "" {TEXT -1 33 "Warning, computation interrupted\n" }}}{PARA 0 "" 0 "" {TEXT -1 85 "No answer, but luckily we raised the information level so that \+ we could read the line" }}{PARA 6 "" 1 "" {TEXT -1 46 "\"Tried to dete rmine the sign of --> 2*Im(1/k)\"" }}{PARA 0 "" 0 "" {TEXT -1 55 "Appa rently we have to put constraints on the parameter " }{XPPEDIT 18 0 "k " "6#%\"kG" }{TEXT -1 1 "." }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 20 "infolevel[int] := 0:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 71 "Integrate(1/((1+k^2*sin(t)^3)^3 * sqrt(1+k^2*sin(t)^2) ),\n t=0..Pi/ 4);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%$IntG6$*&\"\"\"F'*&),&F'F'*&) %\"kG\"\"#F')-%$sinG6#%\"tG\"\"$F'F'F4F',&F'F'*&F,F')F0F.F'F'#F'F.!\" \"/F3;\"\"!,$*&\"\"%F9%#PiGF'F'" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 22 "value(%) assuming k>0;" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#,8*& ,8*(\"$%=\"\"\")%\"kG\"\"#F(,&F+F(*$F)F(F(#F(F+F(**\"#^F()F*\"\")F(F,F .F+F.F(*(\"$k#F()F*\"\"'F(F,F.!\"\"*(\"$U\"F(F1F(F,F.F(*(\"#5F()F*F;F( F,F.F7**\"#QF(FF()F*\"\"%F(F,F. F(**\"#LF(FBF(F,F.F+F.F7**\"\"*F()F*\"#7F(F,F.F+F.F7**\"#))F(F5F(F,F.F +F.F7F(,>*&\"$;#F(FF(F5F(Fdq F(F(*(FXF(FBF(FdqF(F7*(FbqF(F)F(FdqF(F7F(,&F*F(F(F7!\"#,&F*F(F(F(Fjq,& F(F(F-F(FjqF*F7F^qFjq,&-%#lnG6#**,**&F+F(FdqF(F7F[qF(*(F+F(,*F^qF(*$Fd qF(F7*&F)F(FdqF(F(F(F(F.F,F.F(F-F(F(F*F+,(*(F+F(F)F(FdqF(F(*&FCF(F^qF( F7F-F(F(,&*$FBF(F(F2F7F7F7-F_r6#*(,*FfrF7FgrF(F+F(*&F+F(FerF.F(F(F)F(F ^qF(F(F(Fer#F7F+F(F7/F^q-%'RootOfG6#,&*&F)F()%#_ZGF[qF(F(F(F(F(F(*&#F( FgoF(*.F*F+,**&F;F(F5F(F(*&\"#?F(FBF(F7*&F2F(F)F(F(F]qF(F(,&F\\sF(F(F7 F7,**$F5F(F(F\\sF(F-F7F(F7F7F\\rFbs-%*EllipticKG6#*&F*F(F\\rFbsF(F(F7* &#F(FgoF(*.F*F+F^tF(FctF7FdtF7F\\rFbs-%*EllipticFG6$,$*&F+F7F+F.F(FitF (F(F(*&F.F(*,,**$F1F(F(*&F+F(F5F(F7F\\sF(F(F7F(FctF7FdtF7F\\rFbsFftF(F (F(*&#F(F+F(*,FduF(FctF7FdtF7F\\rFbsF]uF(F(F7*&#F(F+F(*,FduF(FctF7FdtF 7F\\rF.-%*EllipticEGFhtF(F(F7*&F.F(*,FduF(FctF7FdtF7F\\rF.-F^vF_uF(F(F (*&F\\rFbs-%$SumG6$*(,0*&,**&FbqF(F1F(F7*&FfqF(F5F(F(*&FXF(FBF(F7*&Fbq F(F)F(F7F(FdqF(F(*&,**&F]qF(F1F(F7*&FXF(F5F(F(F\\sF7*&FbqF(F)F(F(F(F^q F(F(*&FgoF(F1F(F7*&FhpF(F5F(F(*&FhpF(FBF(F(*&F[qF(F)F(F7FgoF7F(,&*.\"# aF()FiqF+F()F[rF+F()F\\rF+F(F)F(F^qF(F7**FiwF(FjwF(F[xF(F\\xF(F7F7-%+E llipticPiG6$,$*&F(F(,&F(F7FfrF(F7F7FitF(FcsF(F(*&F\\rFbs-Fdv6$*(FgvF(F gwF7-F_x6%F`uFaxFitF(FcsF(F7" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 12 "simplify(%);" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#,$*&#\"\"\"\"$3 \"F&*.,bv*,\"$`\"F&)%\"kG\"#5F&,&\"\"#F&*$)F-F0F&F&#F&F0,&F&F&F1F&F3F0 F3!\"\"**\"$;#F&F,F&-%*EllipticEG6$,$*&F0F5F0F3F&*&F-F&F4#F5F0F&F0F3F& *,\"$9\"F&)F-\"#7F&F/F3F4F3F0F3F5**\"#IF&FAF&F/F3F4F3F&**\"$7$F&FAF&F4 F3F0F3F&**\"$3%F&)F-\"\"'F&F4F3F0F3F&*,\"#'*F&)F-\"\"%F&F/F3F4F3F0F3F5 *(-%$sumG6$**,<*&\"#=F&)F-\"\")F&F&*&\"#:F&FIF&F5*&FZF&FMF&F5*&\"\"$F& F2F&F&FVF&*(\"#6F&%'_alphaGF&FWF&F&*(\"#9F&FIF&FjnF&F5*&FMF&FjnF&F&*( \"\"(F&F2F&FjnF&F5*(F_oF&FWF&)FjnF0F&F&*(\"#>F&FIF&FaoF&F5*(F\\oF&FMF& FaoF&F&*(F_oF&F2F&FaoF&F&F&,&-%#lnG6#,$**,**&F0F&FaoF&F&FgnF5*(F0F&,*F jnF&*$FaoF&F5*&F2F&FaoF&F&F&F&F3F/F3F5F1F5F&F-F0,(*(F0F&F2F&FaoF&F&*&F NF&FjnF&F5F1F&F&,&*$FMF&F&FXF5F5F5F5-Fho6#*(,*F`pF5FapF&F0F&*&F0F&F_pF 3F&F&F2F&FjnF&F&F&Fjn!\"#F_pF>/Fjn-%'RootOfG6#,&*&F2F&)%#_ZGFgnF&F&F&F &F&F4F3FIF&F5*(\"#;F&-%$SumG6$*(FTF&-%+EllipticPiG6$,$*&F&F&,&F&F5F`pF &F5F5F=F&,&*&F2F&FjnF&F&F&F&F5F]qF&FMF&F5*(FPF&F4F3FMF&F5**FXF&FgqF&FM F&F0F3F5**FXF&FgqF&FIF&F0F3F5**FXF&-Fhq6$*(FTF&-F\\r6%F;F^rF=F&FarF5F] qF&FMF&F0F3F&**FXF&FgrF&FIF&F0F3F&*(F0F&FgqF&FIF&F5*(F0F&FgrF&FIF&F&*( FfqF&FgrF&FMF&F&**\"#[F&-%*EllipticFGF:F&F,F&F0F3F&**F7F&F8F&)F-F\\oF& F0F3F5**F7F&-F96#F=F&FIF&F0F3F5**F7F&F8F&FWF&F0F3F5**F7F&FgsF&FWF&F0F3 F&**F7F&F8F&FMF&F0F3F&**F7F&F8F&FAF&F0F3F&*(\"#aF&F8F&FesF&F&*(\"$'[F& F8F&FWF&F&*(\"$y$F&FgsF&F,F&F5*(F`tF&FgsF&FWF&F5*(\"$K%F&FbsF&F2F&F5*( FBF&FAF&FbsF&F&*(FetF&FgsF&FMF&F5*(FbtF&FgsF&FIF&F&*(\"$G&F&FbsF&FMF&F 5*(FJF&FbsF&FIF&F5*(FJF&FesF&FbsF&F5*(FetF&F8F&F2F&F&*(FetF&F8F&FMF&F& *(FbtF&FgsF&FAF&F&*(FbtF&F8F&FIF&F5*(F^tF&F8F&)F-FfqF&F5*(FbtF&F8F&F,F &F&*(FbtF&F8F&FAF&F5*(F^tF&FgsF&FesF&F5*(F^tF&FgsF&FbuF&F&**F7F&-%*Ell ipticKGFhsF&FMF&F0F3F&*(FjtF&FhuF&FMF&F&*(FJF&FhuF&FIF&F&*(FJF&FesF&Fh uF&F&*(\"#UF&F,F&FbsF&F5*(FDF&FWF&FbsF&F&*(FBF&FAF&FhuF&F5*(FDF&FWF&Fh uF&F5*(FetF&FgsF&F2F&F5**\"#CF&FAF&FhuF&F0F3F&**F7F&FbsF&FMF&F0F3F5**F 7F&FgsF&FAF&F0F3F5**FdvF&FWF&FhuF&F0F3F5**FasF&F,F&FhuF&F0F3F5**\"$k#F &FIF&FhuF&F0F3F&**F7F&FgsF&FMF&F0F3F5**F7F&F8F&FIF&F0F3F&*(FetF&FhuF&F 2F&F&*(F^vF&F,F&FhuF&F&**FdvF&FAF&FbsF&F0F3F5*(F0F&FgrF&FWF&F&*,FNF&FP F&F4F3FMF&F0F3F5*,FNF&FPF&F4F3F2F&F0F3F5**FXF&FPF&F4F3F2F&F5*(FXF&FPF& F4F3F5*(FfqF&FgqF&F2F&F5**FjvF&FbsF&FIF&F0F3F5**FdvF&FbsF&FWF&F0F3F&** F7F&FgsF&FesF&F0F3F&*(\"$S#F&FIF&F4F3F&*(\"%Y6F&FWF&F4F3F5*(\"$;)F&FMF &F4F3F&**F@F&FIF&F/F3F4F3F5**\"$#zF&FWF&F/F3F4F3F&**\"$E%F&F,F&F/F3F4F 3F5**\"$_&F&FMF&F/F3F4F3F5*,\"#**F&FIF&F/F3F4F3F0F3F&*,\"#FF&FesF&F/F3 F4F3F0F3F&*,FjvF&FWF&F/F3F4F3F0F3F&*(\"$a'F&F,F&F4F3F&*(\"$c\"F&FAF&F4 F3F5*(\"#yF&FesF&F4F3F&**\"$C'F&F,F&F4F3F0F3F5**\"$?\"F&FWF&F4F3F0F3F& **F7F&F,F&FgsF&F0F3F5*(FfqF&FgrF&F2F&F&*(F0F&FgqF&FWF&F5F&,(FfpF&*(FNF &F2F&F0F3F&FXF&F5F-F\\q,&FfpF&F&F5F5,**$FIF&F&FfpF&F1F5F&F5F5F4F>F&F5 " }}}}{SECT 1 {PARA 0 "" 0 "" {TEXT 270 3 "14." }{TEXT -1 9 " Compute \+ " }{XPPEDIT 18 0 "int((x^a - 1)/ln(x),x=0..1);" "6#-%$intG6$*&,&)%\"xG %\"aG\"\"\"F+!\"\"F+-%#lnG6#F)F,/F);\"\"!F+" }{TEXT -1 18 ", for nonne gative " }{XPPEDIT 18 0 "a" "6#%\"aG" }{TEXT -1 2 ".\n" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 8 "restart;" }}}{EXCHG {PARA 0 "> " 0 " " {MPLTEXT 1 0 45 "integral := Integrate((x^a-1)/ln(x), x=0..1);" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#>%)integralG-%$IntG6$*&,&)%\"xG%\"aG\" \"\"F-!\"\"F--%#lnG6#F+F./F+;\"\"!F-" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 23 "value(%) assuming a>=0;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%#lnG6#,&\"\"\"F'%\"aGF'" }}}{PARA 0 "" 0 "" {TEXT -1 99 "We us e a trick to validate the answer: we differentiate the integral with r espect to the parameter " }{XPPEDIT 18 0 "a" "6#%\"aG" }{TEXT -1 147 " , Maple will interchange the differentiation and integration, and the \+ integral obtained can be determined. Afterwards we integrate with resp ect to " }{XPPEDIT 18 0 "a" "6#%\"aG" }{TEXT -1 40 " and determine the integration constant." }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 18 "di ff(integral, a);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%$IntG6$)%\"xG%\" aG/F';\"\"!\"\"\"" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 23 "value( %) assuming a>=0;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#*&\"\"\"F$,&F$F$% \"aGF$!\"\"" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 15 "integrate(%, a);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%#lnG6#,&\"\"\"F'%\"aGF'" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 24 "eval(integral - %, a=0);" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#-%$IntG6$\"\"!/%\"xG;F&\"\"\"" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 9 "value(%);" }}{PARA 11 "" 1 " " {XPPMATH 20 "6#\"\"!" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}}{SECT 1 {PARA 0 "" 0 "" {TEXT 271 3 "15." }{TEXT -1 9 " Compute " }{XPPEDIT 18 0 "Int(ln(x)/((x + a)*(x-1)), x=0..infinity" "6#-%$IntG6$ *&-%#lnG6#%\"xG\"\"\"*&,&F*F+%\"aGF+F+,&F*F+F+!\"\"F+F0/F*;\"\"!%)infi nityG" }{TEXT -1 15 ", for positive " }{XPPEDIT 18 0 "a" "6#%\"aG" } {TEXT -1 2 ".\n" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 8 "restart;" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 46 "Integrate(ln(x)/((x+a)*(x -1)), x=0..infinity);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%$IntG6$*(-% #lnG6#%\"xG\"\"\",&F*F+%\"aGF+!\"\",&F*F+F+F.F./F*;\"\"!%)infinityG" } }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 22 "value(%) assuming a>0;" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#,$*&#\"\"\"\"\"#F&*&,&*$)%#PiGF'F&F&*$ )-%#lnG6#%\"aGF'F&F&F&,&F&F&F2F&!\"\"F&F&" }}}{PARA 0 "" 0 "" {TEXT -1 26 "Two numerical validations:" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 23 "value(eval(%%=%, a=1));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/,$*&\"\"%!\"\"%#PiG\"\"#\"\"\"F$" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 26 "value(eval(%%%=%%, a=12));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/,**&\"#E!\"\"%#PiG\"\"#\"\"\"*&#F)\"#8F**$)-%#lnG6#F)F )F*F*F**&F,F**&F0F*-F16#\"\"$F*F*F**&#F*F&F**$)F5F)F*F*F*,&*&F&F'F(F)F **&F9F**$)-F16#\"#7F)F*F*F*" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 20 "testeq(simplify(%));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%%trueG" } }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}}{SECT 1 {PARA 0 "" 0 "" {TEXT 272 3 "16." }{TEXT -1 9 " Compute " }{XPPEDIT 18 0 "Int(ln(x^ 2+1)/(x^2+1),x);" "6#-%$IntG6$*&-%#lnG6#,&*$%\"xG\"\"#\"\"\"F.F.F.,&*$ F,F-F.F.F.!\"\"F," }{TEXT -1 1 "\n" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 8 "restart;" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 31 "Integrate(ln(x^2+1)/(x^2+1),x);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-% $IntG6$*&-%#lnG6#,&*$)%\"xG\"\"#\"\"\"F/F/F/F/F*!\"\"F-" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 14 "f := value(%);" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#>%\"fG,2*(^##!\"\"\"\"#\"\"\"-%#lnG6#,&%\"xGF+^#F)F+F+- F-6#,&*$)F0F*F+F+F+F+F+F+*&^##F+F*F+-%&dilogG6#*&F'F+,&F0F+^#F+F+F+F+F +*(F8F+F,F+-F-FF+F2F+F+*&F'F+ -F;6#*&F8F+F/F+F+F+*(F'F+FHF+-F-FLF+F+*&^##F)FEF+)FHF*F+F+" }}}{PARA 0 "" 0 "" {TEXT -1 48 "A complicated answer involving complex function s" }}{PARA 0 "" 0 "" {TEXT -1 110 "It may surprise you a bit that the \+ real integrand gives a result full of complex units. Nevertheless, a \+ real " }{XPPEDIT 18 0 "x" "6#%\"xG" }{TEXT -1 58 " will give a real re sult as the following three examples (" }{XPPEDIT 18 0 "x=0,x=1,x=2" " 6%/%\"xG\"\"!/F$\"\"\"/F$\"\"#" }{TEXT -1 11 ") indicate." }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 19 "evalc(subs(x=0,f));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,$*&#\"\"\"\"\"#F&*&%#PiGF&-%#lnG6#F'F&F&!\"\"" }} }{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 19 "evalc(subs(x=1,f));" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#,(*&^##\"\"\"\"\"#F'-%&dilogG6#^$F&#! \"\"F(F'F'*&^#F-F'-F*6#^$F&F&F'F'*&#F'\"\")F'*&%#PiGF'-%#lnG6#F(F'F'F. " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 9 "evalf(%);" }}{PARA 11 " " 1 "" {XPPMATH 20 "6#^$$!+Uflf\"*!#5$\"\"!F(" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 19 "evalc(subs(x=2,f));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,,*&^##\"\"\"\"\"#F'-%&dilogG6#^$F&!\"\"F'F'*&^##F-F(F' -F*6#^$F&F'F'F'*&#F'F(F'*&-%'arctanG6#F&F'-%#lnG6#\"\"&F'F'F-*&F&F'*&F 7F'-F;6##F=\"\"%F'F'F'*&F&F'*&F:F'-F86#F(F'F'F'" }}}{EXCHG {PARA 0 "> \+ " 0 "" {MPLTEXT 1 0 9 "evalf(%);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#^$ $!+$e?Pi&!#5$\"\"!F(" }}}{PARA 0 "" 0 "" {TEXT -1 54 "Let us increase \+ confidence in the following primitive:" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 2 "f;" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#,2*(^##!\"\"\"\" #\"\"\"-%#lnG6#,&%\"xGF)^#F'F)F)-F+6#,&*$)F.F(F)F)F)F)F)F)*&^##F)F(F)- %&dilogG6#*&F%F),&F.F)^#F)F)F)F)F)*(F6F)F*F)-F+F:F)F)*&^##F)\"\"%F))F* F(F)F)*(F6F)-F+6#F " 0 "" {MPLTEXT 1 0 10 "diff(f,x);" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#,:*(^## !\"\"\"\"#\"\"\",&%\"xGF)^#F'F)F'-%#lnG6#,&*$)F+F(F)F)F)F)F)F)**F,F)-F .6#F*F)F+F)F0F'F)*&#F)\"\"%F)*&-F.6#*&F%F),&F+F)^#F)F)F)F),&F)F)*&^##F )F(F)F=F)F)F'F)F)*(FAF)F*F'F:F)F)*(FAF)F4F)F=F'F)*(FAF)F4F)F*F'F)*(FAF )F=F'F-F)F)**-F.6#F=F)F+F)F0F'F>F)F)*&F7F)*&-F.6#*&FAF)F*F)F),&F)F)*&F %F)F*F)F)F'F)F)*(F%F)F=F'FLF)F)*(F%F)FHF)F*F'F)*(F%F)FHF)F=F'F)" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 12 "simplify(%);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#*&-%#lnG6#,&*$)%\"xG\"\"#\"\"\"F,F,F,F,F'!\"\"" }} }}{SECT 1 {PARA 0 "" 0 "" {TEXT 273 3 "17." }{TEXT -1 9 " Compute " } {XPPEDIT 18 0 "Int(x^2/sqrt(x^6+1),x);" "6#-%$IntG6$*&%\"xG\"\"#-%%sqr tG6#,&*$F'\"\"'\"\"\"F/F/!\"\"F'" }{TEXT -1 1 "\n" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 8 "restart;" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 30 "Integrate(x^2/sqrt(x^6+1), x);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%$IntG6$*&%\"xG\"\"#,&*$)F'\"\"'\"\"\"F-F-F-#!\"\"F(F' " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 9 "value(%);" }}{PARA 11 " " 1 "" {XPPMATH 20 "6#,$*&#\"\"\"\"\"$F&-%(arcsinhG6#*$)%\"xGF'F&F&F& " }}}{PARA 0 "" 0 "" {TEXT -1 76 "We could also have obtained this res ult by use of substitution of variables." }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 25 "with(Student[Calculus1]):" }}}{EXCHG {PARA 0 "> " 0 " " {MPLTEXT 1 0 30 "Integrate(x^2/sqrt(x^6+1), x);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%$IntG6$*&%\"xG\"\"#,&*$)F'\"\"'\"\"\"F-F-F-#!\"\"F(F' " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 35 "Rule[change, x=tan(y)^( 1/3), y](%);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%$IntG6$*&%\"xG\"\"# ,&*$)F(\"\"'\"\"\"F.F.F.#!\"\"F)F(-F%6$,$*&#F.\"\"$F.-%$secG6#%\"yGF.F .F:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 33 "subs(sec(y)=sqrt(1+t an(y)^2), %);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%$IntG6$*&%\"xG\"\" #,&*$)F(\"\"'\"\"\"F.F.F.#!\"\"F)F(-F%6$,$*&#F.\"\"$F.*$,&F.F.*$)-%$ta nG6#%\"yGF)F.F.#F.F)F.F.F>" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 14 "value(rhs(%));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,$*&#\"\"\"\"\"$ F&-%(arcsinhG6#-%$tanG6#%\"yGF&F&" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 20 "subs(tan(y)=x^3, %);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,$*&#\"\"\"\"\"$F&-%(arcsinhG6#*$)%\"xGF'F&F&F&" }}}{PARA 0 "" 0 " " {TEXT -1 58 "Let us verify this indefinite integral by differentiati on." }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 10 "diff(%,x);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#*&%\"xG\"\"#,&*$)F$\"\"'\"\"\"F*F*F*#!\"\"F% " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}}{SECT 1 {PARA 0 " " 0 "" {TEXT 274 3 "18." }{TEXT -1 9 " Compute " }{XPPEDIT 18 0 "Int(t an(arctan(x)/3),x)" "6#-%$IntG6$-%$tanG6#*&-%'arctanG6#%\"xG\"\"\"\"\" $!\"\"F-" }{TEXT -1 1 "\n" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 8 " restart;" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 31 "Integrate(tan(a rctan(x)/3), x);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%$IntG6$-%$tanG6# ,$*&#\"\"\"\"\"$F,-%'arctanG6#%\"xGF,F,F1" }}}{PARA 0 "" 0 "" {TEXT -1 85 "No direct integration result as we shall see. We shall use subs titution of variables." }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 9 "val ue(%);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%$intG6$-%$tanG6#,$*&#\"\" \"\"\"$F,-%'arctanG6#%\"xGF,F,F1" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 56 "Student:-Calculus1:-Rule[change, y=1/3*arctan(x), y]( %);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%$IntG6$-%$tanG6#,$*&#\"\"\" \"\"$F--%'arctanG6#%\"xGF-F-F2-F%6$,&*&F.F--F(6#%\"yGF-F-*(F.F-F7F-)-F (6#,$*&F.F-F9F-F-\"\"#F-F-F9" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 14 "value(rhs(%));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,**&#\"\")\"\" *\"\"\"-%#lnG6#-%$cosG6#%\"yGF(!\"\"*&#\"\"%F'F(-F*6#,&*&F3F()F,\"\"#F (F(\"\"$F0F(F(*&#F9F:F(*&F(F(F6F0F(F(*&#F(\"\"'F(*&F(F(*$F8F(F0F(F(" } }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 25 "subs(y=1/3*arctan(x), %); " }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,**&#\"\")\"\"*\"\"\"-%#lnG6#-%$co sG6#,$*&#F(\"\"$F(-%'arctanG6#%\"xGF(F(F(!\"\"*&#\"\"%F'F(-F*6#,&*&F:F ()F,\"\"#F(F(F2F7F(F(*&#F@F2F(*&F(F(F=F7F(F(*&#F(\"\"'F(*&F(F(*$F?F(F7 F(F(" }}}{PARA 0 "" 0 "" {TEXT -1 45 "Let us verify this result by dif ferentiation." }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 10 "diff(%,x); " }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,**&#\"\")\"#F\"\"\"*(-%$sinG6#,$* &#F(\"\"$F(-%'arctanG6#%\"xGF(F(F(,&F(F(*$)F4\"\"#F(F(!\"\"-%$cosGF,F9 F(F(*&#\"#KF'F(**F:F(F*F(F5F9,&*&\"\"%F()F:F8F(F(F0F9F9F(F9*&#\"#;\"\" *F(**F@!\"#F:F(F*F(F5F9F(F(*&#F(FGF(*(F:!\"$F*F(F5F9F(F(" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 10 "normal(%);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#**-%$sinG6#,$*&#\"\"\"\"\"$F*-%'arctanG6#%\"xGF*F*F*,&F *F**$)F/\"\"#F*F*!\"\"-%$cosGF&!\"$,&*&\"\"%F*)F5F3F*F*F+F4!\"#" }}} {PARA 0 "" 0 "" {TEXT -1 17 "We recognize the " }{XPPEDIT 18 0 "tan(ar ctan(x)/3)" "6#-%$tanG6#*&-%'arctanG6#%\"xG\"\"\"\"\"$!\"\"" }{TEXT -1 6 " term." }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 67 "% * cos(1/3* arctan(x)) / sin(1/3*arctan(x)) * \n tan(1/3*arctan(x));" }}{PARA 11 " " 1 "" {XPPMATH 20 "6#**,&\"\"\"F%*$)%\"xG\"\"#F%F%!\"\"-%$cosG6#,$*&# F%\"\"$F%-%'arctanG6#F(F%F%!\"#,&*&\"\"%F%)F+F)F%F%F1F*F5-%$tanGF-F%" }}}{PARA 0 "" 0 "" {TEXT -1 61 "What rests us is to prove that the den ominator is equal to 1." }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 9 "de nom(%);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#*(,&\"\"\"F%*$)%\"xG\"\"#F% F%F%)-%$cosG6#,$*&#F%\"\"$F%-%'arctanG6#F(F%F%F)F%),&*&\"\"%F%F*F%F%F1 !\"\"F)F%" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 18 "f := unapply(% ,x);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"fGf*6#%\"xG6\"6$%)operator G%&arrowGF(*(,&\"\"\"F.*$)9$\"\"#F.F.F.)-%$cosG6#,$*&#F.\"\"$F.-%'arct anG6#F1F.F.F2F.),&*&\"\"%F.F3F.F.F:!\"\"F2F.F(F(F(" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 5 "f(0);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#\" \"\"" }}}{PARA 0 "" 0 "" {TEXT -1 67 "All we have to do now is to show that the derivative is equal to 0." }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 17 "convert(f(x),ln);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6# *(,&\"\"\"F%*$)%\"xG\"\"#F%F%F%)-%%coshG6#,&*&#F%\"\"'F%-%#lnG6#,&F%F% *&^#!\"\"F%F(F%F%F%F%*&#F%F1F%-F36#,&F%F%*&F(F%^#F%F%F%F%F8F)F%),&*&\" \"%F%F*F%F%\"\"$F8F)F%" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 15 "c onvert(%,exp);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#*(,&\"\"\"F%*$)%\"xG \"\"#F%F%F%),&*&#F%F)F%-%$expG6#,&*&#F%\"\"'F%-%#lnG6#,&F%F%*&^#!\"\"F %F(F%F%F%F%*&#F%F4F%-F66#,&F%F%*&F(F%^#F%F%F%F%F;F%F%*&F-F%*&F%F%F.F;F %F%F)F%),&*&\"\"%F%F*F%F%\"\"$F;F)F%" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 10 "diff(%,x);" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#,(**\" \"#\"\"\"%\"xGF&),&*&#F&F%F&-%$expG6#,&*&#F&\"\"'F&-%#lnG6#,&F&F&*&^#! \"\"F&F'F&F&F&F&*&#F&F2F&-F46#,&F&F&*&F'F&^#F&F&F&F&F9F&F&*&F+F&*&F&F& F,F9F&F&F%F&),&*&\"\"%F&F(F&F&\"\"$F9F%F&F&*,F%F&,&F&F&*$)F'F%F&F&F&F) F&FCF&,&*&F+F&*&,&*&^##F9F2F&F6F9F&*&FQF&F>F9F&F&F,F&F&F&*&#F&F%F&*&F, F9FOF&F&F9F&F&*,\"#;F&FIF&)F)FGF&FDF&FLF&F&" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 12 "simplify(%);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#\" \"!" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}}{SECT 1 {PARA 0 "" 0 "" {TEXT 275 3 "19." }{TEXT -1 9 " Compute " }{XPPEDIT 18 0 "in t(arcsin(x/a)^2, x)" "6#-%$intG6$*$-%'arcsinG6#*&%\"xG\"\"\"%\"aG!\"\" \"\"#F+" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 8 "restart;" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 28 "Integrate(arcsin(x/a)^2, x);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%$In tG6$*$)-%'arcsinG6#*&%\"xG\"\"\"%\"aG!\"\"\"\"#F-F," }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 9 "value(%);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%$intG6$*$)-%'arcsinG6#*&%\"xG\"\"\"%\"aG!\"\"\"\"#F-F," }}}{PARA 0 "" 0 "" {TEXT -1 63 "No direct integration result. We use substituti on of variables." }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 54 "Student: -Calculus1:-Rule[change, arcsin(x/a)=y, y](%);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%$IntG6$*$)-%'arcsinG6#*&%\"xG\"\"\"%\"aG!\"\"\"\"#F. F--F%6$*()-F*6#-%$sinG6#%\"yGF1F.-%$cosGF:F.F/F.F;" }}}{PARA 0 "" 0 " " {TEXT -1 116 "Let us assume the generic case and simplify without an y scruples. After all, we shall verify the primitive function." }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 27 "simplify(rhs(%), symbolic); " }}{PARA 11 "" 1 "" {XPPMATH 20 "6#*&%\"aG\"\"\"-%$IntG6$*&)%\"yG\"\" #F%-%$cosG6#F+F%F+F%" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 9 "valu e(%);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#*&%\"aG\"\"\",(*&)%\"yG\"\"#F %-%$sinG6#F)F%F%*&F*F%F+F%!\"\"*(F*F%F)F%-%$cosGF-F%F%F%" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 23 "subs(y=arcsin(x/a), %);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#*&%\"aG\"\"\",(*&)-%'arcsinG6#*&%\"xGF%F$!\" \"\"\"#F%-%$sinG6#F)F%F%*&F/F%F0F%F.*(F/F%F)F%-%$cosGF2F%F%F%" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 12 "simplify(%);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,(*&)-%'arcsinG6#*&%\"xG\"\"\"%\"aG!\"\"\"\"#F+F*F +F+*&F.F+F*F+F-**F.F+F&F+,$*&,&*$)F,F.F+F-*$)F*F.F+F+F+F,!\"#F-#F+F.F, F+F+" }}}{PARA 0 "" 0 "" {TEXT -1 45 "Let us verify this result by dif ferentiation." }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 10 "diff(%,x); " }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,,*,\"\"#\"\"\"-%'arcsinG6#*&%\"xG F&%\"aG!\"\"F&F+F&F,F-,&F&F&*&F+F%F,!\"#F-#F-F%F&*$)F'F%F&F&F%F-*(F%F& F.F1,$*&,&*$)F,F%F&F-*$)F+F%F&F&F&F,F0F-#F&F%F&*,F%F&F'F&F5F1F,F-F+F&F -" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 12 "simplify(%);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#*$)-%'arcsinG6#*&%\"xG\"\"\"%\"aG!\"\"\"\"#F *" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}}{SECT 1 {PARA 0 " " 0 "" {TEXT 276 3 "20." }{TEXT -1 9 " Compute " }{XPPEDIT 18 0 "Int(1 /sqrt(a*sqrt(x)+b),x)" "6#-%$IntG6$*&\"\"\"F'-%%sqrtG6#,&*&%\"aGF'-F)6 #%\"xGF'F'%\"bGF'!\"\"F0" }{TEXT -1 1 "\n" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 8 "restart;" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 33 "Integrate(1/sqrt(a*sqrt(x)+b),x);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6# -%$IntG6$*&\"\"\"F'*$,&*&%\"aGF'%\"xG#F'\"\"#F'%\"bGF'#F'F.!\"\"F," }} }{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 9 "value(%);" }}{PARA 11 "" 1 " " {XPPMATH 20 "6#,$*,\"\"#\"\"\"%\"bG#\"\"$F%%#PiG#!\"\"F%%\"aG!\"#,&* (\"\"%F&F)F,F*#F&F%F&**\"\"'F,F*F2,&**F1F&%\"xGF2F-F&F'F,F&\"\")F,F&,& *(F7F2F-F&F'F,F&F&F&F2F&F&F&" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 13 "radnormal(%);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,$*,\"\"%\"\"\" \"\"$!\"\"%\"bG#F&\"\"#,(*&F+F&F)F&F(*(*(%\"xG#F(F+,&*&F0F&%\"aGF&F&*& F0F*F)F&F&F&F)F(F*F0F*F4F&F(*(F+F&F)F&F/F*F&F&F4!\"#F(" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 20 "simplify(diff(%,x));" }}{PARA 11 " " 1 "" {XPPMATH 20 "6#*&\"\"\"F$*&%\"bG#F$\"\"#*&,&*&%\"aGF$%\"xG#F$F( F$F&F$F$F&!\"\"#F$F(F/" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 29 "s implify(%, assume=positive);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#*&\"\" \"F$*$,&*&%\"aGF$%\"xG#F$\"\"#F$%\"bGF$#F$F+!\"\"" }}}{EXCHG {PARA 0 " > " 0 "" {MPLTEXT 1 0 0 "" }}}}{SECT 1 {PARA 0 "" 0 "" {TEXT 277 3 "21 ." }{TEXT -1 39 " Compute the following infinite sums.\n\n" }{TEXT 343 3 "(a)" }{TEXT -1 4 " " }{XPPEDIT 18 0 "Sum((2*k+3)/(k+1)/(k+2) /(k+3),k=0..infinity)" "6#-%$SumG6$**,&*&\"\"#\"\"\"%\"kGF*F*\"\"$F*F* ,&F+F*F*F*!\"\",&F+F*F)F*F.,&F+F*F,F*F./F+;\"\"!%)infinityG" }{TEXT -1 2 "\n\n" }{TEXT 344 3 "(b)" }{TEXT -1 3 " " }{XPPEDIT 18 0 "Sum(( k^2+k-1)/(k+2)!, k=1..infinity)" "6#-%$SumG6$*&,(*$%\"kG\"\"#\"\"\"F)F +F+!\"\"F+-%*factorialG6#,&F)F+F*F+F,/F);F+%)infinityG" }{TEXT -1 2 " \n\n" }{TEXT 345 3 "(c)" }{TEXT -1 4 " " }{XPPEDIT 18 0 "Sum(k/(k-1 )^2/(k+1)^2,k=2..infinity)" "6#-%$SumG6$*(%\"kG\"\"\"*$,&F'F(F(!\"\"\" \"#F+*$,&F'F(F(F(F,F+/F';F,%)infinityG" }{TEXT -1 2 "\n\n" }{TEXT 346 3 "(d)" }{TEXT -1 4 " " }{XPPEDIT 18 0 "1/4+Sum((3*k+2)/k^3/(k+1)/( k+2),k=1..infinity)" "6#,&*&\"\"\"F%\"\"%!\"\"F%-%$SumG6$**,&*&\"\"$F% %\"kGF%F%\"\"#F%F%*$F/F.F',&F/F%F%F%F',&F/F%F0F%F'/F/;F%%)infinityGF% " }{TEXT -1 2 "\n\n" }{TEXT 347 3 "(e)" }{TEXT -1 4 " " }{XPPEDIT 18 0 "Sum((k^3+7*k^2+4*k+6)/(k^2*(k^2+2)*(k^2+2*k+3)),k=1..infinity)" "6#-%$SumG6$*&,**$%\"kG\"\"$\"\"\"*&\"\"(F+*$F)\"\"#F+F+*&\"\"%F+F)F+F +\"\"'F+F+*(F)F/,&*$F)F/F+F/F+F+,(*$F)F/F+*&F/F+F)F+F+F*F+F+!\"\"/F);F +%)infinityG" }{TEXT -1 1 "\n" }}{SECT 1 {PARA 5 "" 0 "" {TEXT 348 3 " (a)" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 8 "restart;" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 48 "Sum((2*k+3)/((k+1)*(k+2)*(k+3)), k= 0..infinity);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%$SumG6$**,&*&\"\"# \"\"\"%\"kGF*F*\"\"$F*F*,&F+F*F*F*!\"\",&F+F*F)F*F.,&F+F*F,F*F./F+;\" \"!%)infinityG" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 20 "evalf(%); # numeric" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"++++]7!\"*" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 18 "value(%%); # exact" }}{PARA 11 "" 1 "" {XPPMATH 20 "6##\"\"&\"\"%" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}}{SECT 1 {PARA 5 "" 0 "" {TEXT 349 3 "(b)" }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 8 "restart;" }}}{EXCHG {PARA 0 " > " 0 "" {MPLTEXT 1 0 37 "Sum((k^2+k-1)/(k+2)!, k=1..infinity);" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#-%$SumG6$*&,(*$)%\"kG\"\"#\"\"\"F,F*F, F,!\"\"F,-%*factorialG6#,&F*F,F+F,F-/F*;F,%)infinityG" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 20 "evalf(%); # numeric" }}{PARA 11 " " 1 "" {XPPMATH 20 "6#$\"+++++]!#5" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 18 "value(%%); # exact" }}{PARA 11 "" 1 "" {XPPMATH 20 "6 ##\"\"\"\"\"#" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}} {SECT 1 {PARA 5 "" 0 "" {TEXT 350 3 "(c)" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 8 "restart;" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 38 "Sum(k/(k-1)^2/(k+1)^2, k=2..infinity);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%$SumG6$*(%\"kG\"\"\",&F'F(F(!\"\"!\"#,&F'F(F(F(F+/F';\"\"#%)in finityG" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 20 "evalf(%); # num eric" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"++++DJ!#5" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 18 "value(%%); # exact" }}{PARA 11 "" 1 "" {XPPMATH 20 "6##\"\"&\"#;" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 9 "evalf(%);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"++++DJ!#5" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}}{SECT 1 {PARA 5 "" 0 "" {TEXT 351 3 "(d)" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 8 "restart;" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 52 "1/4 + Sum((3*k+2)/(k^3*(k+1) *(k+2)), k=1..infinity);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,&#\"\"\" \"\"%F%-%$SumG6$**,&*&\"\"$F%%\"kGF%F%\"\"#F%F%F.!\"$,&F.F%F%F%!\"\",& F.F%F/F%F2/F.;F%%)infinityGF%" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 20 "evalf(%); # numeric" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"+.p0- 7!\"*" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 18 "value(%%); # exact " }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%%ZetaG6#\"\"$" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 9 "evalf(%);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"+.p0-7!\"*" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}} {SECT 1 {PARA 5 "" 0 "" {TEXT 352 3 "(e)" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 8 "restart;" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 66 "Sum((k^3+7*k^2+4*k+6)/(k^2*(k^2+2)*(k^2+2*k+3)),\n k=1..infinity);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%$SumG6$**,**$)%\"kG\"\"$\"\"\"F,*& \"\"(F,)F*\"\"#F,F,*&\"\"%F,F*F,F,\"\"'F,F,F*!\"#,&*$F/F,F,F0F,!\"\",( F6F,*&F0F,F*F,F,F+F,F7/F*;F,%)infinityG" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 20 "evalf(%); # numeric" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"+M2g68!\"*" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 18 "value(% %); # exact" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,&#\"\"\"\"\"$!\"\"*&\" \"'F'%#PiG\"\"#F%" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 9 "evalf(% );" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"+N2g68!\"*" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}}}{SECT 1 {PARA 0 "" 0 "" {TEXT 278 3 "22." }{TEXT -1 9 " Compute " }{XPPEDIT 18 0 "Sum((-3)^k * MATRIX([[ 2*n],[2*k]]),k=0..n);" "6#-%$SumG6$*&),$\"\"$!\"\"%\"kG\"\"\"-%'MATRIX G6#7$7#*&\"\"#F,%\"nGF,7#*&F3F,F+F,F,/F+;\"\"!F4" }{TEXT -1 1 "\n" }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 8 "restart;" }}}{EXCHG {PARA 0 " > " 0 "" {MPLTEXT 1 0 38 "Sum((-3)^k*binomial(2*n,2*k), k=0..n);" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#-%$SumG6$*&)!\"$%\"kG\"\"\"-%)binomial G6$,$*&\"\"#F*%\"nGF*F*,$*&F0F*F)F*F*F*/F);\"\"!F1" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 9 "value(%);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6 #,$*&#\"\"\"\"\"%F&**%#PiG#F&\"\"#)!\"$F%F&-%*LegendrePG6%,&%\"nGF&#F& F+!\"\"F*#F4F+F&)F',&F2F&F*F&F&F&F&" }}}{PARA 0 "" 0 "" {TEXT -1 101 " Let us try to find a simpler answer. From a number of calculations we \+ shall conjecture a new formula." }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 35 "evalc(simplify([seq(%, n=0..15)]));" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#72\"\"\"!\"#!\")\"#k!$G\"!$7&\"%'4%!%#>)!&oF$\"'W@E!')G C&!(_r4#\");sx;!)KWbL!*Gx@M\"\"+C=ut5" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 11 "ifactor(%);" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#72\"\" \",$-%!G6#\"\"#!\"\",$*$)F&\"\"$F$F**$)F&\"\"'F$,$*$)F&\"\"(F$F*,$*$)F &\"\"*F$F**$)F&\"#7F$,$*$)F&\"#8F$F*,$*$)F&\"#:F$F**$)F&\"#=F$,$*$)F& \"#>F$F*,$*$)F&\"#@F$F**$)F&\"#CF$,$*$)F&\"#DF$F*,$*$)F&\"#FF$F**$)F& \"#IF$" }}}{PARA 0 "" 0 "" {TEXT -1 21 "We guess the formula " } {XPPEDIT 18 0 "4^n*cos(2*Pi*n/3)" "6#*&)\"\"%%\"nG\"\"\"-%$cosG6#**\" \"#F'%#PiGF'F&F'\"\"$!\"\"F'" }{TEXT -1 24 " and verify a few cases." }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 43 "[seq(ifactor(4^n*cos(2*Pi* n/3)), n=0..15)];" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#72\"\"\",$-%!G6# \"\"#!\"\",$*$)F&\"\"$F$F**$)F&\"\"'F$,$*$)F&\"\"(F$F*,$*$)F&\"\"*F$F* *$)F&\"#7F$,$*$)F&\"#8F$F*,$*$)F&\"#:F$F**$)F&\"#=F$,$*$)F&\"#>F$F*,$* $)F&\"#@F$F**$)F&\"#CF$,$*$)F&\"#DF$F*,$*$)F&\"#FF$F**$)F&\"#IF$" }}} {PARA 0 "" 0 "" {TEXT -1 60 "The solution can also be expressed in a h ypergeometric term:" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 33 "hyper geom([-n, -n+1/2],[1/2],-3);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%*hyp ergeomG6%7$,$%\"nG!\"\",&F(F)#\"\"\"\"\"#F,7#F+!\"$" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 28 "simplify([seq(%, n=0..15)]);" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#72\"\"\"!\"#!\")\"#k!$G\"!$7&\"%'4%!%#>)!&oF$\" 'W@E!')GC&!(_r4#\");sx;!)KWbL!*Gx@M\"\"+C=ut5" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}}{SECT 1 {PARA 0 "" 0 "" {TEXT 279 3 "23." } {TEXT -1 9 " Compute " }{XPPEDIT 18 0 "Sum(MATRIX([[2*k],[k]])*(1/2)^( 2*k),k=0..n)" "6#-%$SumG6$*&-%'MATRIXG6#7$7#*&\"\"#\"\"\"%\"kGF.7#F/F. )*&F.F.F-!\"\"*&F-F.F/F.F./F/;\"\"!%\"nG" }{TEXT -1 1 "\n" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 8 "restart;" }}}{EXCHG {PARA 0 "> " 0 " " {MPLTEXT 1 0 41 "Sum(binomial(2*k,k)*(1/2)^(2*k), k=0..n);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%$SumG6$*&-%)binomialG6$,$*&\"\"#\"\"\"%\"k GF-F-F.F-)#F-F,F*F-/F.;\"\"!%\"nG" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 9 "value(%);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,$**\"\"# \"\"\",&%\"nGF&F&F&F&-%)binomialG6$,&*&F%F&F(F&F&F%F&F'F&)#F&F%F,F&F& " }}}{PARA 0 "" 0 "" {TEXT -1 21 "Numerical validation:" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 84 "\{seq(testeq( \n value(subs( n=i,%%)) = eval(subs(n=i,%))\n ),\n i=0..15)\};" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#<#%%trueG" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}}{SECT 1 {PARA 0 "" 0 "" {TEXT 280 3 "24." }{TEXT -1 51 " Co mpute the product of the first 32 prime numbers." }}{EXCHG {PARA 0 "> \+ " 0 "" {MPLTEXT 1 0 8 "restart;" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 30 "Product(ithprime(i), i=1..32);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%(ProductG6$-%)ithprimeG6#%\"iG/F);\"\"\"\"#K" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 9 "value(%);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#\"TIEXn4h+H\">Tsqzr8xSxi_!zk*e_" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 11 "ifactor(%);" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#*\\o-% !G6#\"\"#\"\"\"-F%6#\"\"$F(-F%6#\"\"&F(-F%6#\"\"(F(-F%6#\"#6F(-F%6#\"# 8F(-F%6#\"#F(-F%6#\"#BF(-F%6#\"#HF(-F%6#\"#JF(-F%6#\"#PF(- F%6#\"#TF(-F%6#\"#VF(-F%6#\"#ZF(-F%6#\"#`F(-F%6#\"#fF(-F%6#\"#hF(-F%6# \"#nF(-F%6#\"#rF(-F%6#\"#tF(-F%6#\"#zF(-F%6#\"#$)F(-F%6#\"#*)F(-F%6#\" #(*F(-F%6#\"$,\"F(-F%6#\"$.\"F(-F%6#\"$2\"F(-F%6#\"$4\"F(-F%6#\"$8\"F( -F%6#\"$F\"F(-F%6#\"$J\"F(" }}}{PARA 0 "" 0 "" {TEXT -1 41 "Alternativ ely you could do the following:" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 28 "[seq(ithprime(i), i=1..32)];" }}{PARA 12 "" 1 "" {XPPMATH 20 " 6#7B\"\"#\"\"$\"\"&\"\"(\"#6\"#8\"#<\"#>\"#B\"#H\"#J\"#P\"#T\"#V\"#Z\" #`\"#f\"#h\"#n\"#r\"#t\"#z\"#$)\"#*)\"#(*\"$,\"\"$.\"\"$2\"\"$4\"\"$8 \"\"$F\"\"$J\"" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 15 "convert(% ,`*`);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#\"TIEXn4h+H\">Tsqzr8xSxi_!zk *e_" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 11 "ifactor(%);" }} {PARA 12 "" 1 "" {XPPMATH 20 "6#*\\o-%!G6#\"\"#\"\"\"-F%6#\"\"$F(-F%6# \"\"&F(-F%6#\"\"(F(-F%6#\"#6F(-F%6#\"#8F(-F%6#\"#F(-F%6#\" #BF(-F%6#\"#HF(-F%6#\"#JF(-F%6#\"#PF(-F%6#\"#TF(-F%6#\"#VF(-F%6#\"#ZF( -F%6#\"#`F(-F%6#\"#fF(-F%6#\"#hF(-F%6#\"#nF(-F%6#\"#rF(-F%6#\"#tF(-F%6 #\"#zF(-F%6#\"#$)F(-F%6#\"#*)F(-F%6#\"#(*F(-F%6#\"$,\"F(-F%6#\"$.\"F(- F%6#\"$2\"F(-F%6#\"$4\"F(-F%6#\"$8\"F(-F%6#\"$F\"F(-F%6#\"$J\"F(" }}} {PARA 0 "" 0 "" {TEXT -1 11 "Or even do:" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 31 "`*`(seq(ithprime(i), i=1..32));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#\"TIEXn4h+H\">Tsqzr8xSxi_!zk*e_" }}}{EXCHG {PARA 0 "> \+ " 0 "" {MPLTEXT 1 0 0 "" }}}}{SECT 1 {PARA 0 "" 0 "" {TEXT 281 3 "25. " }{TEXT -1 21 " Compute the product " }{XPPEDIT 18 0 "Product(1-1/k^2 ,k=2..infinity)" "6#-%(ProductG6$,&\"\"\"F'*&F'F'*$%\"kG\"\"#!\"\"F,/F *;F+%)infinityG" }{TEXT -1 3 ". \n" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 8 "restart;" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 32 "Product(1-1/k^2, k=2..infinity);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#- %(ProductG6$,&\"\"\"F'*&F'F'*$)%\"kG\"\"#F'!\"\"F-/F+;F,%)infinityG" } }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 9 "value(%);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6##\"\"\"\"\"#" }}}{PARA 0 "" 0 "" {TEXT -1 62 "You ca n also find first a closed formula for the product with " }{XPPEDIT 18 0 "k" "6#%\"kG" }{TEXT -1 19 " running from 2 to " }{XPPEDIT 18 0 " N" "6#%\"NG" }{TEXT -1 23 " and then compute with " }{TEXT 0 5 "limit " }{TEXT -1 15 " the limit for " }{XPPEDIT 18 0 "N" "6#%\"NG" }{TEXT -1 19 " going to infinity." }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 25 "Product(1-1/k^2, k=2..N);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%(Pr oductG6$,&\"\"\"F'*&F'F'*$)%\"kG\"\"#F'!\"\"F-/F+;F,%\"NG" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 9 "value(%);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,$*&#\"\"\"\"\"#F&*(-%&GAMMAG6#%\"NGF&-F*6#,&F,F&F'F&F& -F*6#,&F,F&F&F&!\"#F&F&" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 21 " Limit(%, N=infinity);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%&LimitG6$,$ *&#\"\"\"\"\"#F)*(-%&GAMMAG6#%\"NGF)-F-6#,&F/F)F*F)F)-F-6#,&F/F)F)F)! \"#F)F)/F/%)infinityG" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 9 "val ue(%);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6##\"\"\"\"\"#" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}}{SECT 1 {PARA 0 "" 0 "" {TEXT 282 3 "26." }{TEXT -1 9 " Compute " }{XPPEDIT 18 0 "Sum((m*k-1), k=1.. n)" "6#-%$SumG6$,&*&%\"mG\"\"\"%\"kGF)F)F)!\"\"/F*;F)%\"nG" }{TEXT -1 69 " and compare the result with Formula 0.122 of Gradshteyn and Ryzhi k.\n" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 8 "restart;" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 21 "Sum((m*k-1), k=1..n);" }}{PARA 11 " " 1 "" {XPPMATH 20 "6#-%$SumG6$,&*&%\"mG\"\"\"%\"kGF)F)F)!\"\"/F*;F)% \"nG" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 9 "value(%);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,(*(\"\"#!\"\"%\"mG\"\"\",&%\"nGF(F(F(F%F(*( F%F&F'F(F)F(F&F*F&" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 10 "facto r(%);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,$*(\"\"#!\"\"%\"nG\"\"\",(*& %\"mGF(F'F(F(F+F(F%F&F(F(" }}}{PARA 0 "" 0 "" {TEXT -1 76 "This is the correct result; Formula 0.122 of Gradshteyn and Ryzhik is wrong." }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}}{SECT 1 {PARA 0 "" 0 " " {TEXT 283 3 "27." }{TEXT -1 9 " Compute " }{XPPEDIT 18 0 "Sum(k^2*x^ 2,k=1..n-1)" "6#-%$SumG6$*&%\"kG\"\"#%\"xGF(/F';\"\"\",&%\"nGF,F,!\"\" " }{TEXT -1 69 " and compare the result with Formula 0.114 of Gradshte yn and Ryzhik.\n" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 8 "restart; " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 23 "Sum(k^2*x^2, k=1..n-1); " }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%$SumG6$*&)%\"kG\"\"#\"\"\")%\"xG F)F*/F(;F*,&%\"nGF*F*!\"\"" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 9 "value(%);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,(*&#\"\"\"\"\"$F&*&)% \"xG\"\"#F&)%\"nGF'F&F&F&*&#F&F+F&*&)F-F+F&F)F&F&!\"\"*&#F&\"\"'F&*&F) F&F-F&F&F&" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 10 "factor(%);" } }{PARA 11 "" 1 "" {XPPMATH 20 "6#,$*,\"\"'!\"\"%\"xG\"\"#%\"nG\"\"\",& *&F(F*F)F*F*F*F&F*,&F)F*F*F&F*F*" }}}{PARA 0 "" 0 "" {TEXT -1 75 "This is the correct result; Formula 0.114 of Gradshteyn and Ryzhik is wron g" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}}}{MARK "0" 0 } {VIEWOPTS 1 1 0 1 1 1803 1 1 1 1 }{PAGENUMBERS 0 1 2 33 1 1 }