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{SECT 0 {PARA 3 "" 0 "" {TEXT -1 49 "Integration pre-requisites for La place Transforms" }}{PARA 0 "" 0 "" {TEXT -1 37 "by Peter Stone, Nanai mo, B.C., Canada" }}{PARA 0 "" 0 "" {TEXT -1 19 "Version: 27.3.2007" }}{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 20 "Integration by parts" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{PARA 0 "" 0 "" {TEXT -1 3 "If " }{XPPEDIT 18 0 "f(x) = u(x)*`.`*v(x)" "6#/-%\"fG6#%\"xG*(-%\"uG6#F' \"\"\"%\".GF,-%\"vG6#F'F," }{TEXT -1 8 ", where " }{XPPEDIT 18 0 "u(x) " "6#-%\"uG6#%\"xG" }{TEXT -1 5 " and " }{XPPEDIT 18 0 "v(x)" "6#-%\"v G6#%\"xG" }{TEXT -1 75 " are two differentiable functions, the product rule for differentiation is:" }}{PARA 256 "" 0 "" {TEXT -1 4 " f '" } {XPPEDIT 18 0 "``(x)" "6#-%!G6#%\"xG" }{TEXT -1 5 " = u'" }{XPPEDIT 18 0 "``(x)" "6#-%!G6#%\"xG" }{TEXT -1 1 " " }{TEXT 259 1 "." }{TEXT -1 1 " " }{XPPEDIT 18 0 "v(x) + u(x)" "6#,&-%\"vG6#%\"xG\"\"\"-%\"uG6# F'F(" }{TEXT -1 1 " " }{TEXT 264 1 "." }{TEXT -1 3 " v'" }{XPPEDIT 18 0 "``(x)" "6#-%!G6#%\"xG" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 2 "or" }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "d/dx;" "6#*& %\"dG\"\"\"%#dxG!\"\"" }{TEXT -1 3 " [ " }{XPPEDIT 18 0 "u*v" "6#*&%\" uG\"\"\"%\"vGF%" }{TEXT -1 5 " ] = " }{TEXT 265 1 "u" }{TEXT -1 1 " " }{XPPEDIT 18 0 "dv/dx;" "6#*&%#dvG\"\"\"%#dxG!\"\"" }{TEXT -1 3 " + " }{TEXT 266 1 "v" }{TEXT -1 1 " " }{XPPEDIT 18 0 "du/dx;" "6#*&%#duG\" \"\"%#dxG!\"\"" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 13 "For exa mple, " }{XPPEDIT 18 0 "d/dx;" "6#*&%\"dG\"\"\"%#dxG!\"\"" }{TEXT -1 2 "[ " }{XPPEDIT 18 0 "x^2*sin(x);" "6#*&%\"xG\"\"#-%$sinG6#F$\"\"\"" }{TEXT -1 5 " ] = " }{XPPEDIT 18 0 "2*x*sin(x);" "6#*(\"\"#\"\"\"%\"xG F%-%$sinG6#F&F%" }{TEXT -1 3 " + " }{XPPEDIT 18 0 "x^2*cos(x);" "6#*&% \"xG\"\"#-%$cosG6#F$\"\"\"" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 19 "diff(x^2*sin(x),x);" }}{PARA 11 "" 1 "" {XPPMATH 20 " 6#,&*&%\"xG\"\"\"-%$sinG6#F%F&\"\"#*&)F%F*F&-%$cosGF)F&F&" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 24 "Integrating the formula " }{XPPEDIT 18 0 "d/dx;" "6#*&%\"dG\"\"\"%#dxG!\"\"" }{TEXT -1 3 " [ " }{XPPEDIT 18 0 "u*v" "6#*&%\"uG\"\"\"%\"vGF%" }{TEXT -1 5 " ] = " }{TEXT 267 1 "u" }{TEXT -1 1 " " }{XPPEDIT 18 0 "dv/dx;" "6#*&%#dvG\"\"\"%#dxG!\"\"" } {TEXT -1 3 " + " }{TEXT 268 1 "v" }{TEXT -1 1 " " }{XPPEDIT 18 0 "du/d x;" "6#*&%#duG\"\"\"%#dxG!\"\"" }{TEXT -1 18 ", with respect to " } {TEXT 269 1 "x" }{TEXT -1 7 " gives:" }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "u*v =Int(u*``(dv/dx),x) +Int(v*``(du/dx),x)" "6#/*& %\"uG\"\"\"%\"vGF&,&-%$IntG6$*&F%F&-%!G6#*&%#dvGF&%#dxG!\"\"F&%\"xGF&- F*6$*&F'F&-F.6#*&%#duGF&F2F3F&F4F&" }{TEXT -1 2 ". " }}{PARA 0 "" 0 " " {TEXT -1 37 "Re-arranging this equation we obtain:" }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "Int(u*``(dv/dx),x)=u*v - Int(v*``( du/dx),x)" "6#/-%$IntG6$*&%\"uG\"\"\"-%!G6#*&%#dvGF)%#dxG!\"\"F)%\"xG, &*&F(F)%\"vGF)F)-F%6$*&F4F)-F+6#*&%#duGF)F/F0F)F1F0" }{TEXT -1 2 ", " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{TEXT 263 18 "__________________" }{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 13 "which is the " }{TEXT 260 28 "integration by parts formula" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}} {SECT 1 {PARA 4 "" 0 "" {TEXT -1 10 "Example 1 " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "Int(x*cos(x),x);" "6#-%$IntG6$*&%\"xG\" \"\"-%$cosG6#F'F(F'" }{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 0 "" } }{PARA 0 "" 0 "" {TEXT -1 4 "Let " }{XPPEDIT 18 0 "u = x" "6#/%\"uG%\" xG" }{TEXT -1 5 " and " }{XPPEDIT 18 0 "v = sin(x)" "6#/%\"vG-%$sinG6# %\"xG" }{TEXT -1 43 " in the integration by parts formula. Then " } {XPPEDIT 18 0 "du/dx = 1;" "6#/*&%#duG\"\"\"%#dxG!\"\"F&" }{TEXT -1 5 " and " }{XPPEDIT 18 0 "dv/dx = cos(x);" "6#/*&%#dvG\"\"\"%#dxG!\"\"-% $cosG6#%\"xG" }{TEXT -1 5 " so: " }}{PARA 256 "" 0 "" {TEXT -1 1 " " } {XPPEDIT 18 0 "Int(x*cos(x),x) =Int(u*``(dv/dx),x)" "6#/-%$IntG6$*&%\" xG\"\"\"-%$cosG6#F(F)F(-F%6$*&%\"uGF)-%!G6#*&%#dvGF)%#dxG!\"\"F)F(" }} {PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "``= u*v-Int(v*``(du/d x),x)" "6#/%!G,&*&%\"uG\"\"\"%\"vGF(F(-%$IntG6$*&F)F(-F$6#*&%#duGF(%#d xG!\"\"F(%\"xGF3" }{TEXT -1 1 " " }}{PARA 256 "" 0 "" {TEXT -1 1 " " } {XPPEDIT 18 0 "``=x*sin(x)-Int(sin(x),x)" "6#/%!G,&*&%\"xG\"\"\"-%$sin G6#F'F(F(-%$IntG6$-F*6#F'F'!\"\"" }{TEXT -1 1 " " }}{PARA 256 "" 0 "" {XPPEDIT 18 0 "`` = x*sin(x)+cos(x)+c;" "6#/%!G,(*&%\"xG\"\"\"-%$sinG6 #F'F(F(-%$cosG6#F'F(%\"cGF(" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 26 "Int(x*cos(x),x);\n value(%);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%$IntG6$*&%\"xG\"\"\"-%$ cosG6#F'F(F'" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,&-%$cosG6#%\"xG\"\"\" *&F'F(-%$sinGF&F(F(" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }} }{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 4 "The " } {TEXT 0 7 "student" }{TEXT -1 30 " package contains a procedure " } {TEXT 0 8 "intparts" }{TEXT -1 88 ", which applies the integration by \+ parts formula to a specified integral. The procedure " }{TEXT 0 8 "int parts" }{TEXT -1 123 " takes the integral it is to be applied to as it s first argument, and the second argument is the expression to be take n as " }{TEXT 301 1 "u" }{TEXT -1 17 " in the formula. " }}{PARA 0 "" 0 "" {TEXT -1 90 "The procedure intparts takes two arguments. The firs t argument is an integral of the form " }{TEXT 303 15 "Int(u*dv/dx, x) " }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 34 "The second argument \+ is the factor " }{TEXT 302 1 "u" }{TEXT -1 16 " in the formula " } {XPPEDIT 18 0 "Int(u*``(dv/dx),x) = u*v-Int(v*``(du/dx),x)" "6#/-%$Int G6$*&%\"uG\"\"\"-%!G6#*&%#dvGF)%#dxG!\"\"F)%\"xG,&*&F(F)%\"vGF)F)-F%6$ *&F4F)-F+6#*&%#duGF)F/F0F)F1F0" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 25 "We can use the procedure " }{TEXT 0 8 "intparts" }{TEXT -1 58 " to provide the intermediate step in the previous example." }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 53 "with(student):\nintparts(Int(x*cos(x),x),x);\nvalue(%);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,&*&%\"xG\"\"\"-%$sinG6#F%F&F&-%$IntG6$F'F%! \"\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,&*&%\"xG\"\"\"-%$sinG6#F%F&F& -%$cosGF)F&" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 10 "Example 2 " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "Int(x^2*sin(x) ,x);" "6#-%$IntG6$*&%\"xG\"\"#-%$sinG6#F'\"\"\"F'" }{TEXT -1 1 " " }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 47 "Applying \+ the integration by parts formula with " }{XPPEDIT 18 0 "u = x^2;" "6#/ %\"uG*$%\"xG\"\"#" }{TEXT -1 5 " and " }{XPPEDIT 18 0 "v = - cos(x)" "6#/%\"vG,$-%$cosG6#%\"xG!\"\"" }{TEXT -1 12 ", (so that " }{XPPEDIT 18 0 "du/dx = 2*x;" "6#/*&%#duG\"\"\"%#dxG!\"\"*&\"\"#F&%\"xGF&" } {TEXT -1 5 " and " }{XPPEDIT 18 0 "dv/dx = sin(x);" "6#/*&%#dvG\"\"\"% #dxG!\"\"-%$sinG6#%\"xG" }{TEXT -1 10 " ) gives: " }}{PARA 256 "" 0 " " {TEXT -1 1 " " }{XPPEDIT 18 0 "Int(x^2*sin(x),x)=Int(u*``(dv/dx),x) " "6#/-%$IntG6$*&%\"xG\"\"#-%$sinG6#F(\"\"\"F(-F%6$*&%\"uGF--%!G6#*&%# dvGF-%#dxG!\"\"F-F(" }{TEXT -1 1 " " }}{PARA 256 "" 0 "" {TEXT -1 1 " \+ " }{XPPEDIT 18 0 "`` = u*v-Int(v*``(du/dx),x)" "6#/%!G,&*&%\"uG\"\"\"% \"vGF(F(-%$IntG6$*&F)F(-F$6#*&%#duGF(%#dxG!\"\"F(%\"xGF3" }{TEXT -1 1 " " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "``=-x^2*cos(x)- Int(-cos(x)*``(2*x),x)" "6#/%!G,&*&%\"xG\"\"#-%$cosG6#F'\"\"\"!\"\"-%$ IntG6$,$*&-F*6#F'F,-F$6#*&F(F,F'F,F,F-F'F-" }{TEXT -1 1 " " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "`` = -x^2*cos(x)+2*Int(x*co s(x),x);" "6#/%!G,&*&%\"xG\"\"#-%$cosG6#F'\"\"\"!\"\"*&F(F,-%$IntG6$*& F'F,-F*6#F'F,F'F,F," }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 38 "T he first example provides the result " }{XPPEDIT 18 0 "Int(x*cos(x),x) = x*sin(x)+cos(x)+c;" "6#/-%$IntG6$*&%\"xG\"\"\"-%$cosG6#F(F)F(,(*&F( F)-%$sinG6#F(F)F)-F+6#F(F)%\"cGF)" }{TEXT -1 37 ", so, making use of t his, we obtain: " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "I nt(x^2*sin(x),x) = -x^2*cos(x)+2*x*sin(x)+2*cos(x)+c[1]" "6#/-%$IntG6$ *&%\"xG\"\"#-%$sinG6#F(\"\"\"F(,**&F(F)-%$cosG6#F(F-!\"\"*(F)F-F(F--F+ 6#F(F-F-*&F)F--F16#F(F-F-&%\"cG6#F-F-" }{TEXT -1 2 ", " }}{PARA 0 "" 0 "" {TEXT -1 6 "where " }{XPPEDIT 18 0 "c[1] = 2*c;" "6#/&%\"cG6#\"\" \"*&\"\"#F'F%F'" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {PARA 0 "" 0 "" {TEXT 260 4 "Note" }{TEXT -1 16 ": The procedure " } {TEXT 0 8 "intparts" }{TEXT -1 93 " is applied to all integrals in an \+ expression, but does not affect other factors or summands." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 88 "stud ent[intparts](Int(x^2*sin(x),x),x^2);\nsimplify(%);\nstudent[intparts] (%,x);\nvalue(%);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,&*&)%\"xG\"\"#\" \"\"-%$cosG6#F&F(!\"\"-%$IntG6$,$*&F&F(F)F(!\"#F&F," }}{PARA 11 "" 1 " " {XPPMATH 20 "6#,&*&)%\"xG\"\"#\"\"\"-%$cosG6#F&F(!\"\"*&F'F(-%$IntG6 $*&F&F(F)F(F&F(F(" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,(*&)%\"xG\"\"#\" \"\"-%$cosG6#F&F(!\"\"*(F'F(F&F(-%$sinGF+F(F(*&F'F(-%$IntG6$F.F&F(F," }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,(*&)%\"xG\"\"#\"\"\"-%$cosG6#F&F(! \"\"*(F'F(F&F(-%$sinGF+F(F(*&F'F(F)F(F(" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 10 "Example 3 " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "Int(x*exp(2*x),x);" "6#-%$IntG6$*&%\"xG\"\"\"-%$expG6# *&\"\"#F(F'F(F(F'" }{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {PARA 0 "" 0 "" {TEXT -1 47 "Applying the integration by parts formula with " }{XPPEDIT 18 0 "u = x;" "6#/%\"uG%\"xG" }{TEXT -1 5 " and " } {XPPEDIT 18 0 "v = 1/2;" "6#/%\"vG*&\"\"\"F&\"\"#!\"\"" }{TEXT -1 1 " \+ " }{XPPEDIT 18 0 "exp(2*x)" "6#-%$expG6#*&\"\"#\"\"\"%\"xGF(" }{TEXT -1 12 ", (so that " }{XPPEDIT 18 0 "du/dx = 1;" "6#/*&%#duG\"\"\"%#dx G!\"\"F&" }{TEXT -1 5 " and " }{XPPEDIT 18 0 "dv/dx = exp(2*x);" "6#/* &%#dvG\"\"\"%#dxG!\"\"-%$expG6#*&\"\"#F&%\"xGF&" }{TEXT -1 10 " ) give s: " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "Int(x*exp(2*x) ,x) = Int(u*``(dv/dx),x);" "6#/-%$IntG6$*&%\"xG\"\"\"-%$expG6#*&\"\"#F )F(F)F)F(-F%6$*&%\"uGF)-%!G6#*&%#dvGF)%#dxG!\"\"F)F(" }{TEXT -1 1 " " }}{PARA 256 "" 0 "" {TEXT -1 2 " " }{XPPEDIT 18 0 "`` = u*v-Int(v*``( du/dx),x)" "6#/%!G,&*&%\"uG\"\"\"%\"vGF(F(-%$IntG6$*&F)F(-F$6#*&%#duGF (%#dxG!\"\"F(%\"xGF3" }{TEXT -1 1 " " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "``= x/2 exp(2*x) - Int(exp(2*x)/2,x)" "6#/%!G,&*(% \"xG\"\"\"\"\"#!\"\"-%$expG6#*&F)F(F'F(F(F(-%$IntG6$*&-F,6#*&F)F(F'F(F (F)F*F'F*" }{TEXT -1 1 " " }}{PARA 256 "" 0 "" {TEXT -1 1 " " } {XPPEDIT 18 0 "``=x*exp(2*x)/2-exp(2*x)/4+c" "6#/%!G,(*(%\"xG\"\"\"-%$ expG6#*&\"\"#F(F'F(F(F-!\"\"F(*&-F*6#*&F-F(F'F(F(\"\"%F.F.%\"cGF(" } {TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 49 "student[intparts](Int(x*exp(2*x),x),x);\nvalue(% );" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,&*&#\"\"\"\"\"#F&*&%\"xGF&-%$ex pG6#,$*&F'F&F)F&F&F&F&F&-%$IntG6$,$*&F%F&F*F&F&F)!\"\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,&*&#\"\"\"\"\"#F&*&%\"xGF&-%$expG6#,$*&F'F&F)F&F& F&F&F&*&#F&\"\"%F&F*F&!\"\"" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{SECT 1 {PARA 4 "" 0 " " {TEXT -1 10 "Example 4 " }}{PARA 256 "" 0 "" {TEXT -1 1 " " } {XPPEDIT 18 0 "Int(ln(x),x);" "6#-%$IntG6$-%#lnG6#%\"xGF)" }{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 47 "A pplying the integration by parts formula with " }{XPPEDIT 18 0 "u = ln (x);" "6#/%\"uG-%#lnG6#%\"xG" }{TEXT -1 5 " and " }{XPPEDIT 18 0 "v = \+ x;" "6#/%\"vG%\"xG" }{TEXT -1 12 ", (so that " }{XPPEDIT 18 0 "du/dx \+ = 1/x;" "6#/*&%#duG\"\"\"%#dxG!\"\"*&F&F&%\"xGF(" }{TEXT -1 5 " and " }{XPPEDIT 18 0 "dv/dx = 1;" "6#/*&%#dvG\"\"\"%#dxG!\"\"F&" }{TEXT -1 10 " ) gives: " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "Int (ln(x),x) = Int(u*``(dv/dx),x);" "6#/-%$IntG6$-%#lnG6#%\"xGF*-F%6$*&% \"uG\"\"\"-%!G6#*&%#dvGF/%#dxG!\"\"F/F*" }{TEXT -1 1 " " }}{PARA 256 " " 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "`` = u*v-Int(v*``(du/dx),x)" "6# /%!G,&*&%\"uG\"\"\"%\"vGF(F(-%$IntG6$*&F)F(-F$6#*&%#duGF(%#dxG!\"\"F(% \"xGF3" }{TEXT -1 1 " " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "`` = x*ln(x)-Int(x*`.`*``(1/x),x);" "6#/%!G,&*&%\"xG\"\"\"-%#lnG 6#F'F(F(-%$IntG6$*(F'F(%\".GF(-F$6#*&F(F(F'!\"\"F(F'F4" }{TEXT -1 1 " \+ " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "``=x*ln(x)-Int(1, x)" "6#/%!G,&*&%\"xG\"\"\"-%#lnG6#F'F(F(-%$IntG6$F(F'!\"\"" }{TEXT -1 1 " " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "`` = x*ln(x)- x+c;" "6#/%!G,(*&%\"xG\"\"\"-%#lnG6#F'F(F(F'!\"\"%\"cGF(" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 48 "student[intparts](Int(ln(x),x),ln(x));\nvalue(%);" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#,&*&-%#lnG6#%\"xG\"\"\"F(F)F)-%$IntG6$ F)F(!\"\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,&*&-%#lnG6#%\"xG\"\"\"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 18 "Improper integrals" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{PARA 0 "" 0 "" {TEXT -1 31 "W e consider graphs of the form " }{XPPEDIT 18 0 "y = f(x)" "6#/%\"yG-% \"fG6#%\"xG" }{TEXT -1 8 ", where " }{XPPEDIT 18 0 "f(x)" "6#-%\"fG6#% \"xG" }{TEXT -1 18 " is positive, and " }{XPPEDIT 18 0 "f(x) -> 0" "6# f*6#-%\"fG6#%\"xG7\"6$%)operatorG%&arrowG6\"\"\"!F-F-F-" }{TEXT -1 4 " as " }{XPPEDIT 18 0 "x -> infinity" "6#f*6#%\"xG7\"6$%)operatorG%&arr owG6\"%)infinityGF*F*F*" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 26 "This means that the graph " }{XPPEDIT 18 0 "y = f(x)" "6#/%\"yG-% \"fG6#%\"xG" }{TEXT -1 16 " approaches the " }{TEXT 270 1 "x" }{TEXT -1 12 " axis as an " }{TEXT 260 9 "asymptote" }{TEXT -1 1 "." }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{SECT 1 {PARA 4 "" 0 " " {TEXT -1 9 "Example 1" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 "; " }}}{PARA 0 "" 0 "" {TEXT -1 23 "Consider the function " }{XPPEDIT 18 0 "f(x) = 1/(x^2);" "6#/-%\"fG6#%\"xG*&\"\"\"F)*$F'\"\"#!\"\"" } {TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 44 "The area of the region \+ bounded by the graph " }{XPPEDIT 18 0 "y=1/x^2" "6#/%\"yG*&\"\"\"F&*$% \"xG\"\"#!\"\"" }{TEXT -1 5 " the " }{TEXT 291 1 "x" }{TEXT -1 29 " ax is and the vertical lines " }{XPPEDIT 18 0 "x = 1" "6#/%\"xG\"\"\"" } {TEXT -1 5 " and " }{XPPEDIT 18 0 "x = 4" "6#/%\"xG\"\"%" }{TEXT -1 41 " is illustrated in the following picture." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 329 "pts := [[1, 0],op(op(1,op(1,plot(1/x^2,x=1..4,adaptive=false,numpoints=30)))),[4,0 ]]:\np1 := plots[polygonplot](pts,color=COLOR(RGB,.8,.75,1),style=PATC HNOGRID):\np2 := plot([[[1,0],[1,1]],[[4,0],[4,1/16]]],style=LINE,colo r=black):\np3 := plot(1/x^2,x=0.7..5):\nplots[display]([p1,p2,p3],view =[0..5,0..2],labels=[x,y],ytickmarks=3);\n\n" }}{PARA 13 "" 1 "" {GLPLOT2D 400 297 297 {PLOTDATA 2 "6)-%)POLYGONSG6%7B7$$\"\"\"\"\"!$F* F*7$F(F(7$$\"+XTB36!\"*$\"+['4@9)!#57$$\"+'43C?\"F0$\"+Tml;pF37$$\"+_l J38F0$\"+`=!RAVF 37$$\"+;F0$\"+;l?8QF37$$\"+44E@F0$\"+d0#F37$$\"+'>^=C#F0$\"+hYp*)>F37$$\"+@/R\\BF0$\"+&)Qr6=F37$$\"+! RCIX#F0$\"+7q'=m\"F37$$\"+jR8ZDF0$\"+AGLT:F37$$\"+[$R!fEF0$\"+%3HVT\"F 37$$\"+()y$Qv#F0$\"+J4j=8F37$$\"+C66kGF0$\"++l/>7F37$$\"+D;shHF0$\"+Cv ,S6F37$$\"+9e\")oIF0$\"+>#Q=1\"F37$$\"+3[zqJF0$\"+u'\\j%**!#67$$\"+!>* >xKF0$\"+Iq&4J*Fbr7$$\"+G;\"\\P$F0$\"+!H4'z()Fbr7$$\"+$p2.[$F0$\"+sb!f D)Fbr7$$\"+%e%y*e$F0$\"+Xo-gxFbr7$$\"+RX3&o$F0$\"+TE&QO(Fbr7$$\"+,6,)y $F0$\"+>J6ppFbr7$$\"+7VM%*QF0$\"+VAt$f'Fbr7$$\"\"%F*$\"++++]iFbr7$FgtF +-%&COLORG6&%$RGBG$\"\")!\"\"$\"#v!\"#F)-%&STYLEG6#%,PATCHNOGRIDG-%'CU RVESG6%7$F'F,-%'COLOURG6&F_uF*F*F*-Fgu6#%%LINEG-F[v6%7$F[u7$Fgt$\"3+++ ++++]i!#>F^vFav-F[v6$7Y7$$\"3a**************p!#=$\"3H71`Ej\"3/#!#<7$$ \"37m;H2%>VB(Faw$\"3%\\cj#)R`2\">Fdw7$$\"3oKLe9)Q'ouFaw$\"3g8?+y!RFz\" Fdw7$$\"3B**\\(=AeHq(Faw$\"3'))*Gue)H`o\"Fdw7$$\"3zlm;HwFPzFaw$\"3+9d% )4@H(e\"Fdw7$$\"3'))\\PfvP]M)Faw$\"3)=w3xXjfV\"Fdw7$$\"3/L$3F)yz_()Faw $\"31.oJ#*H,U(*Faw7$$\"3sm;H!H: $f5Fdw$\"39.#*))fMZ6*)Faw7$$\"3=L3_@#)>^6Fdw$\"3i28QjUqXvFaw7$$\"3mmTg gaQO7Fdw$\"3fc$pn1CfC8Fdw$\"3Qv*=FX([*p&Faw7$$\"3ymT 5^^\"eT\"Fdw$\"3]ffhmUp))\\Faw7$$\"34+voNeu1:Fdw$\"3f/t%3=PZS%Faw7$$\" 3ZLLe@#z-g\"Fdw$\"3$Qkt,(p)[!RFaw7$$\"3am;zyMm#o\"Fdw$\"3p0FZ@h(=`$Faw 7$$\"3'****\\(R'3ax\"Fdw$\"3#GA^3]7D<$Faw7$$\"3-++D@Y`o=Fdw$\"3K]>:8G; kGFaw7$$\"3-++Dg(y#e>Fdw$\"3t7WRY+m2EFaw7$$\"3WmT5O]xR?Fdw$\"3MJ;.7?X. CFaw7$$\"3TLLe!*>oO@Fdw$\"3S)*=E6QQ!>#Faw7$$\"3DLLL#*[x=AFdw$\"3#[Ir'3 uHJ?Faw7$$\"3<+v$HXoUJ#Fdw$\"3GdU-$G@r'=Faw7$$\"3ELL$[V'z)R#Fdw$\"3H1# o3p`yt\"Faw7$$\"3)**\\Pz-P:\\#Fdw$\"3b/O=fy)3h\"Faw7$$\"3y*\\7Q+[)zDFd w$\"3ljD)3f!\\-:Faw7$$\"3mL$3FC\"*>n#Fdw$\"3Q;r;65l+9Faw7$$\"3WL3-+vgc FFdw$\"3LMa!zb#)fJ\"Faw7$$\"3Km;zkv(y%GFdw$\"3@-ng-Q)HB\"Faw7$$\"3UmT5 6:oUHFdw$\"3ew7BDy\"[:\"Faw7$$\"3m*\\i]i3_-$Fdw$\"3omI,23n#4\"Faw7$$\" 3Lm;/p,M9JFdw$\"3zSdiN>-J5Faw7$$\"3'*****\\vOOQFdw$\"37f0[\"[_Xz'Fjv7$$\"3*G$3FEw!H#RFdw$\"3 #[aD\\yh!)\\'Fjv7$$\"3M+voGGP8SFdw$\"34=5%Fd w$\"33(oR5pkN%fFjv7$$\"3q*\\P%[TT%>%Fdw$\"3mXJ!*RL/%o&Fjv7$$\"3%ommc=% f$G%Fdw$\"33m>ro]$)\\aFjv7$$\"3cm;HaOzuVFdw$\"3,PC7aE)\\A&Fjv7$$\"3IL3 FXzBlWFdw$\"3.i;5z,Y:]Fjv7$$\"3=++]boM[XFdw$\"3dT\">0TYQ$[Fjv7$$\"38L$ ea/*fVYFdw$\"3*QC/--rvj%Fjv7$$\"30nmmRPzGZFdw$\"3gyZ%Fjv7$$\"3b+ v$H0H'>[Fdw$\"34FOM-i*\\I%Fjv7$$\"39+D1Q+d1\\Fdw$\"3I1$y8#[y`TFjv7$$\" \"&F*$\"3y+++++++SFjv-F_v6&F_u$\"#5FbuF+F+-%*AXESTICKSG6$%(DEFAULTG\" \"$-%+AXESLABELSG6%%\"xG%\"yG-%%FONTG6#Fjhl-%%VIEWG6$;F+F_hl;F+$\"\"#F *" 1 2 0 1 10 0 2 9 1 4 2 1.000000 45.000000 45.000000 0 0 "Curve 1" " Curve 2" "Curve 3" "Curve 4" }}}}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {PARA 0 "" 0 "" {TEXT -1 56 "The area of the region shown is given by \+ the integral: " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "In t(1/x^2,x=1..4)=-1/x" "6#/-%$IntG6$*&\"\"\"F(*$%\"xG\"\"#!\"\"/F*;F(\" \"%,$*&F(F(F*F,F," }{TEXT -1 1 " " }{XPPEDIT 18 0 "PIECEWISE([4, ``],[ ``, ``],[1, ``]);" "6#-%*PIECEWISEG6%7$\"\"%%!G7$F(F(7$\"\"\"F(" } {XPPEDIT 18 0 "``=``(-1/4)-``(-1)" "6#/%!G,&-F$6#,$*&\"\"\"F*\"\"%!\" \"F,F*-F$6#,$F*F,F," }{XPPEDIT 18 0 "``= 1-1/4" "6#/%!G,&\"\"\"F&*&F&F &\"\"%!\"\"F)" }{XPPEDIT 18 0 "``=3/4" "6#/%!G*&\"\"$\"\"\"\"\"%!\"\" " }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 28 "Int(1/x^2,x=1..4);\nvalue(%);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%$IntG6$*&\"\"\"F'*$)%\"xG\"\"#F'!\"\"/F*;F'\"\"% " }}{PARA 11 "" 1 "" {XPPMATH 20 "6##\"\"$\"\"%" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 37 "Similarly, the area under the graph " }{XPPEDIT 18 0 "y = 1/(x^2);" "6#/%\"yG*&\"\"\"F&*$%\"xG \"\"#!\"\"" }{TEXT -1 6 " from " }{XPPEDIT 18 0 "x = 1" "6#/%\"xG\"\" \"" }{TEXT -1 4 " to " }{XPPEDIT 18 0 "x = 5" "6#/%\"xG\"\"&" }{TEXT -1 14 " is given by: " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "Int(1/(x^2),x = 1 .. 5) = -1/x;" "6#/-%$IntG6$*&\"\"\"F(*$%\"xG \"\"#!\"\"/F*;F(\"\"&,$*&F(F(F*F,F," }{TEXT -1 1 " " }{XPPEDIT 18 0 "P IECEWISE([5, ``],[``, ``],[1, ``]);" "6#-%*PIECEWISEG6%7$\"\"&%!G7$F(F (7$\"\"\"F(" }{XPPEDIT 18 0 "`` = ``(-1/5)-``(-1);" "6#/%!G,&-F$6#,$*& \"\"\"F*\"\"&!\"\"F,F*-F$6#,$F*F,F," }{XPPEDIT 18 0 "`` = 1-1/5;" "6#/ %!G,&\"\"\"F&*&F&F&\"\"&!\"\"F)" }{XPPEDIT 18 0 "`` = 4/5;" "6#/%!G*& \"\"%\"\"\"\"\"&!\"\"" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 0 " " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 28 "Int(1/x^2,x=1..5);\nvalu e(%);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%$IntG6$*&\"\"\"F'*$)%\"xG\" \"#F'!\"\"/F*;F'\"\"&" }}{PARA 11 "" 1 "" {XPPMATH 20 "6##\"\"%\"\"&" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 30 "The area under the graph \+ from " }{XPPEDIT 18 0 "x = 1" "6#/%\"xG\"\"\"" }{TEXT -1 4 " to " } {XPPEDIT 18 0 "x = 6" "6#/%\"xG\"\"'" }{TEXT -1 5 " is: " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "Int(1/(x^2),x = 1 .. 6) = -1/x; " "6#/-%$IntG6$*&\"\"\"F(*$%\"xG\"\"#!\"\"/F*;F(\"\"',$*&F(F(F*F,F," } {TEXT -1 1 " " }{XPPEDIT 18 0 "PIECEWISE([6, ``],[``, ``],[1, ``]);" " 6#-%*PIECEWISEG6%7$\"\"'%!G7$F(F(7$\"\"\"F(" }{XPPEDIT 18 0 "`` = ``(- 1/6)-``(-1);" "6#/%!G,&-F$6#,$*&\"\"\"F*\"\"'!\"\"F,F*-F$6#,$F*F,F," } {XPPEDIT 18 0 "`` = 1-1/6;" "6#/%!G,&\"\"\"F&*&F&F&\"\"'!\"\"F)" } {XPPEDIT 18 0 "`` = 5/6;" "6#/%!G*&\"\"&\"\"\"\"\"'!\"\"" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" } }{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 28 "Int(1/x^2,x=1..6);\nvalue(% );" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%$IntG6$*&\"\"\"F'*$)%\"xG\"\"# F'!\"\"/F*;F'\"\"'" }}{PARA 11 "" 1 "" {XPPMATH 20 "6##\"\"&\"\"'" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{GLPLOT2D 368 243 243 {PLOTDATA 2 "6/-%)POLYGONSG6%7B7$$\"\"\"\"\"!$F*F*7$F(F(7$$\"+XTB36!\" *$\"+['4@9)!#57$$\"+'43C?\"F0$\"+Tml;pF37$$\"+_lJ38F0$\"+`=!RAVF37$$\"+;F0$\"+;l?8 QF37$$\"+44E@F0$\" +d0#F37$$\"+'>^=C #F0$\"+hYp*)>F37$$\"+@/R\\BF0$\"+&)Qr6=F37$$\"+!RCIX#F0$\"+7q'=m\"F37$ $\"+jR8ZDF0$\"+AGLT:F37$$\"+[$R!fEF0$\"+%3HVT\"F37$$\"+()y$Qv#F0$\"+J4 j=8F37$$\"+C66kGF0$\"++l/>7F37$$\"+D;shHF0$\"+Cv,S6F37$$\"+9e\")oIF0$ \"+>#Q=1\"F37$$\"+3[zqJF0$\"+u'\\j%**!#67$$\"+!>*>xKF0$\"+Iq&4J*Fbr7$$ \"+G;\"\\P$F0$\"+!H4'z()Fbr7$$\"+$p2.[$F0$\"+sb!fD)Fbr7$$\"+%e%y*e$F0$ \"+Xo-gxFbr7$$\"+RX3&o$F0$\"+TE&QO(Fbr7$$\"+,6,)y$F0$\"+>J6ppFbr7$$\"+ 7VM%*QF0$\"+VAt$f'Fbr7$$\"\"%F*$\"++++]iFbr7$FgtF+-%&COLORG6&%$RGBG$\" \")!\"\"$\"#v!\"#F)-%&STYLEG6#%,PATCHNOGRIDG-%'CURVESG6%7$F'F,-%'COLOU RG6&F_uF*F*F*-Fgu6#%%LINEG-F[v6%7$F[u7$Fgt$\"3+++++++]i!#>F^vFav-F[v6% 7$7$$!3!***************HFjvF(7$F+F(F^vFav-F[v6$7Y7$$\"3U+++++++!)!#=$ \"3++++++]i:!#<7$$\"3=++v=,()G#)Fhw$\"3o(e_D'Gzw9F[x7$$\"3#*****\\P-ud %)Fhw$\"3u!39(e*[zR\"F[x7$$\"3m***\\iN5mo)Fhw$\"3SV[^!fa_K\"F[x7$$\"3^ +++v/[:*)Fhw$\"3]qF0vk3e7F[x7$$\"3a+]73yv8$*Fhw$\"39d9t$y*y_6F[x7$$\"3 c++DT^.7(*Fhw$\"3w&\\s@wz,1\"F[x7$$\"36+volR*f,\"F[x$\"3'*[2#*\\Wj(o*F hw7$$\"31++Dc]i\\'Fhw7$$\"3?+]7!Q6RK\"F[x$\"3ya>B#R[`q&Fhw7$$\"3#***\\7>l159F[x$ \"3#))[_\\^`%H]Fhw7$$\"3G+]7(Go\"*\\\"F[x$\"3StLg:sP\\WFhw7$$\"3'***\\ i/V)ze\"F[x$\"3mTl6tzelRFhw7$$\"3C++]&\\U$z;F[x$\"3E:G!\\&)eea$Fhw7$$ \"33++vR3\")fF[x$\"3qe$*=U2J`EFhw7$$\"3`++]jk,H?F[x$\"3gL.I8v+HC Fhw7$$\"39+]7xuh3@F[x$\"31%[gJPx!\\AFhw7$$\"36++]\"yqK?#F[x$\"3;uQS4f) *f?Fhw7$$\"3?+++YXX$G#F[x$\"3M!R&\\UF&y\">Fhw7$$\"39+]7EtswBF[x$\"3'>c /4Ax-x\"Fhw7$$\"3U+++T&*GfCF[x$\"3-\"H#pN/T`;Fhw7$$\"3K+]7wL()\\DF[x$ \"3=DYg\"zA!Q:Fhw7$$\"3C+]P318OEF[x$\"3kSt.bx,R9Fhw7$$\"3J++D\")48EFF[ x$\"3ONVsG3dX8Fhw7$$\"3!3+D'=%z(3GF[x$\"3e/+7E*[vE\"Fhw7$$\"3/++vBp#z* GF[x$\"3o>X%fZi2>\"Fhw7$$\"3K+]7Fh_!*HF[x$\"3D4&oveJF[x$\"3IH5gb&)e-5Fhw7$$\"3_ +++(*G8[KF[x$\"33*y33)HMy%*Fjv7$$\"3()***\\-\"=7OLF[x$\"3GlNJ)4h\\)*)F jv7$$\"3^+]73cD@MF[x$\"33bb.&)[OV&)Fjv7$$\"32++vv@y:NF[x$\"3W[.2(zS,4) Fjv7$$\"3C+++,&=2g$F[x$\"3b$fY0JqHr(Fjv7$$\"3K++]dhS\"p$F[x$\"3d!>[6q_ 'QtFjv7$$\"3_+]7`EetPF[x$\"3!*Q_x$Q3D-(Fjv7$$\"3%4++&oKUjQF[x$\"31D6U[ 4q*p'Fjv7$$\"3s**\\7'Gcz%RF[x$\"3cre4fm'eT'Fjv7$$\"39+]iYwJOSF[x$\"3Jg &Fjv7$$\"3C+]7sUnxWF[x$\"3W/s@8*Qw)\\Fjv7$$\"3Q+++ " 0 " " {MPLTEXT 1 0 67 "Int(1/x^2,x=1..R);\nexpand(value(%));\nLimit(%,R=in finity);\nvalue(%);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%$IntG6$*&\"\" \"F'*$)%\"xG\"\"#F'!\"\"/F*;F'%\"RG" }}{PARA 11 "" 1 "" {XPPMATH 20 "6 #,&*&\"\"\"F%%\"RG!\"\"F'F%F%" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%&Li mitG6$,&*&\"\"\"F(%\"RG!\"\"F*F(F(/F)%)infinityG" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#\"\"\"" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 31 "The va lue of this limit is the " }{TEXT 260 17 "improper integral" }{TEXT -1 2 " " }{XPPEDIT 18 0 "Int(1/(x^2),x = 0 .. infinity);" "6#-%$IntG6 $*&\"\"\"F'*$%\"xG\"\"#!\"\"/F);\"\"!%)infinityG" }{TEXT -1 11 ", that is, " }}{PARA 256 "" 0 "" {TEXT -1 2 " " }{XPPEDIT 18 0 "Int(1/(x^2) ,x = 0 .. infinity)= Limit(Int(1/(x^2),x = 0 .. R),R = infinity)" "6#/ -%$IntG6$*&\"\"\"F(*$%\"xG\"\"#!\"\"/F*;\"\"!%)infinityG-%&LimitG6$-F% 6$*&F(F(*$F*F+F,/F*;F/%\"RG/F:F0" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 69 "Since the value of the limit is 1, we say that the improp er integral " }{TEXT 260 12 "converges to" }{TEXT -1 4 " 1. " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 47 "Maple can evalu ate improper integrals directly." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 35 "Int(1/x^2,x=1..infinity);\nv alue(%);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%$IntG6$*&\"\"\"F'*$)%\"x G\"\"#F'!\"\"/F*;F'%)infinityG" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#\"\" \"" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 9 "Example 2" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 32 "Consider the improper int egral: " }{XPPEDIT 18 0 "Int(exp(-x),x = 0 .. infinity);" "6#-%$IntG6$ -%$expG6#,$%\"xG!\"\"/F*;\"\"!%)infinityG" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 40 "As in the previous example the graph of " } {XPPEDIT 18 0 "f(x)=exp(-x)" "6#/-%\"fG6#%\"xG-%$expG6#,$F'!\"\"" } {TEXT -1 16 " approaches the " }{TEXT 292 1 "x" }{TEXT -1 25 " axis as an asymptote as " }{XPPEDIT 18 0 "x->infinity" "6#f*6#%\"xG7\"6$%)ope ratorG%&arrowG6\"%)infinityGF*F*F*" }{TEXT -1 92 ". The improper integ ral corresponds to the area of the region enclosed between the graph o f " }{XPPEDIT 18 0 "f(x)=exp(-x)" "6#/-%\"fG6#%\"xG-%$expG6#,$F'!\"\" " }{TEXT -1 6 ", the " }{TEXT 293 1 "x" }{TEXT -1 29 " axis and extend ing from the " }{TEXT 294 1 "y" }{TEXT -1 94 " axis indefinitely to th e right. As in the previous example the area of this region as finite. " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 256 "" 0 "" {TEXT -1 2 " " } {GLPLOT2D 300 221 221 {PLOTDATA 2 "6,-%)POLYGONSG6%7B7$$\"\"!F)F(7$F($ \"\"\"F)7$$\"+[9M#3\"!#5$\"+iW'F0$\"+D*eEQ&F07$$\"+$44E@(F0$\"+B*)Qh[F07$$\"+5!HgE)F0$\"+ y5LvVF07$$\"+(fqgJ*F0$\"+#[,#RRF07$$\"+VnhR5!\"*$\"+<]@$F07$$\"+'>^=C\"Fhn$\"+m+\\))GF07$$\"+@/R\\8Fhn$\"+jL)Rf# F07$$\"+!RCIX\"Fhn$\"+x#>'QBF07$$\"+jR8Z:Fhn$\"+p:dG@F07$$\"+[$R!f;Fhn $\"+`s@.>F07$$\"+()y$Qv\"Fhn$\"+pH3JFhn$\"+U7;19F07$$\"+9e\")o?Fhn$\"+wHNj7F07$$\"+3[zq@Fhn$\"+P!p 39\"F07$$\"+!>*>xAFhn$\"+p3rD5F07$$\"+G;\"\\P#Fhn$\"+Y4F-$*!#67$$\"+$p 2.[#Fhn$\"+Miur$)F\\s7$$\"+%e%y*e#Fhn$\"+O-i.vF\\s7$$\"+RX3&o#Fhn$\"+* oU:#oF\\s7$$\"+,6,)y#Fhn$\"+0'\\V:'F\\s7$$\"+7VM%*GFhn$\"+\"fHN`&F\\s7 $$\"\"$F)$\"+Poqy\\F\\s7$FgtF(-%&COLORG6&%$RGBG$\"\")!\"\"$\"#v!\"#F,- %&STYLEG6#%,PATCHNOGRIDG-%'CURVESG6$7SF*7$$\"3Hmmmm;')=()!#>$\"3_W!*f8 @/l\"*!#=7$$\"3RLLLe'40j\"Fdv$\"3]Z2&R*)ya\\)Fdv7$$\"3mmmm;6m$[#Fdv$\" 3Qb1\"p'Hu+yFdv7$$\"3fmmm;yYULFdv$\"3-#yQ3&4xerFdv7$$\"3%HLL$eF>(>%Fdv $\"3k!o!GYHJslFdv7$$\"3Qmmm\">K'*)\\Fdv$\"3g\"RCl\"RlsXV#Fdv7$$\"3!** ***\\s]k,:Fjy$\"3:?152OUs+SFjy$\"3#)*pZ::'=w9 Fdv7$$\"3immmTc-)*>Fjy$\"3=!f\"Hvu-c8Fdv7$$\"3Mmm;f`@'3#Fjy$\"3Ze^PT8c T7Fdv7$$\"3y****\\nZ)H;#Fjy$\"3K?\\gbT\")\\6Fdv7$$\"3YmmmJy*eC#Fjy$\"3 wm$[&zZKe5Fdv7$$\"3')******R^bJBFjy$\"3AS&RthbWr*Fav7$$\"3f*****\\5a`T #Fjy$\"3y0WC5*pN$*)Fav7$$\"3o****\\7RV'\\#Fjy$\"3zr^!pMCyB)Fav7$$\"3k* ****\\@fke#Fjy$\"3&)>cDuShGvFav7$$\"3/LLL`4NnEFjy$\"3pz$GFjy$\"3ot%z5*= f*)eFav7$$\"3$*******pfaFav 7$$\"\"%F)$\"3A=M()))QcJ=Fav-%'COLOURG6&F_uF)F)F)-F[v6%7$F[u7$Fgt$\"3W %R'yOoqy\\FavFael-Fgu6#%%LINEG-%%TEXTG6$7$Fgt$!\"(FeuQ\"R6\"-F^fl6$7$F ]el$!\"$FeuQ\"xFdfl-F^fl6$7$Fafl$\"$<\"FeuQ\"yFdfl-F^fl6$7$FaflF+Q\"1F dfl-%*AXESTICKSG6$F)F)-%+AXESLABELSG6%%\"xG%\"yG-%%FONTG6#%(DEFAULTG-% %VIEWG6$;FaflF]el;Fafl$\"#7Fbu" 1 2 0 1 10 0 2 9 1 4 2 1.000000 46.000000 45.000000 0 0 "Curve 1" "Curve 2" "Curve 3" "Curve 4" "Curve 5" "Curve 6" "Curve 7" }}{TEXT -1 1 " " }}{PARA 256 "" 0 "" {TEXT -1 2 " " }}{PARA 0 "" 0 "" {TEXT -1 6 "Since " }{XPPEDIT 18 0 "Int(exp(- x),x) = -exp(-x)+c;" "6#/-%$IntG6$-%$expG6#,$%\"xG!\"\"F+,&-F(6#,$F+F, F,%\"cG\"\"\"" }{TEXT -1 28 ", the area under the graph " }{XPPEDIT 18 0 "y = exp(-x);" "6#/%\"yG-%$expG6#,$%\"xG!\"\"" }{TEXT -1 6 " from " }{XPPEDIT 18 0 "x = 0" "6#/%\"xG\"\"!" }{TEXT -1 4 " to " } {XPPEDIT 18 0 "x = R" "6#/%\"xG%\"RG" }{TEXT -1 4 " is:" }}{PARA 256 " " 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "Int(exp(-x),x = 0 .. R);" "6#-%$ IntG6$-%$expG6#,$%\"xG!\"\"/F*;\"\"!%\"RG" }{TEXT -1 3 " = " } {XPPEDIT 18 0 "-exp(-R)+exp(0);" "6#,&-%$expG6#,$%\"RG!\"\"F)-F%6#\"\" !\"\"\"" }{TEXT -1 3 " = " }{XPPEDIT 18 0 "1-exp(-R);" "6#,&\"\"\"F$-% $expG6#,$%\"RG!\"\"F*" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 92 "As the upper limit of the integral tends to infinity, the value of th e integral tends to 1. " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "Int(exp(-x),x = 0 .. infinity)=Limit(Int(exp(-x),x = 0 .. R),R=i nfinity)" "6#/-%$IntG6$-%$expG6#,$%\"xG!\"\"/F+;\"\"!%)infinityG-%&Lim itG6$-F%6$-F(6#,$F+F,/F+;F/%\"RG/F;F0" }{TEXT -1 1 " " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "``=Limit(1-exp(-R),R=infinity)" "6 #/%!G-%&LimitG6$,&\"\"\"F)-%$expG6#,$%\"RG!\"\"F//F.%)infinityG" } {TEXT -1 1 " " }}{PARA 256 "" 0 "" {TEXT -1 2 " " }{XPPEDIT 18 0 "``= 1" "6#/%!G\"\"\"" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 69 "Int(exp(-x),x=0..R);\nvalue( %);\nLimit(1-exp(-R),R=infinity);\nvalue(%);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%$IntG6$-%$expG6#,$%\"xG!\"\"/F*;\"\"!%\"RG" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,&-%$expG6#,$%\"RG!\"\"F)\"\"\"F*" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%&LimitG6$,&-%$expG6#,$%\"RG!\"\"F,\"\"\"F- /F+%)infinityG" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#\"\"\"" }}}{PARA 0 " " 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 50 "Maple can evaluate the improper integral directly." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 37 "Int(exp(-x),x=0..infinity); \nvalue(%);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%$IntG6$-%$expG6#,$%\" xG!\"\"/F*;\"\"!%)infinityG" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#\"\"\" " }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 9 "Example 3" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{PARA 0 "" 0 "" {TEXT -1 32 "Consider the improper integral: " }{XPPEDIT 18 0 "Int(1/( 1+x^2),x = 0 .. infinity);" "6#-%$IntG6$*&\"\"\"F',&F'F'*$%\"xG\"\"#F' !\"\"/F*;\"\"!%)infinityG" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 20 "First we calculate " }{XPPEDIT 18 0 "Int(1/(1+x^2),x = 0 .. 4);" "6#-%$IntG6$*&\"\"\"F',&F'F'*$%\"xG\"\"#F'!\"\"/F*;\"\"!\"\"%" }{TEXT -1 55 ", which is the area of the region bounded by the curve " } {XPPEDIT 18 0 "y=1/(1+x^2)" "6#/%\"yG*&\"\"\"F&,&F&F&*$%\"xG\"\"#F&!\" \"" }{TEXT -1 6 ", the " }{TEXT 295 1 "x" }{TEXT -1 11 " axis, the " } {TEXT 296 1 "y" }{TEXT -1 19 " axis and the line " }{XPPEDIT 18 0 "x = 4;" "6#/%\"xG\"\"%" }{TEXT -1 3 ". " }}{PARA 0 "" 0 "" {TEXT -1 0 " " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 321 "pts :=[[0,0],op(op(1,op (1,plot(1/(1+x^2),x=0..4,adaptive=false,\n nu mpoints=30)))),[4,0]]:\np1 := plots[polygonplot](pts,color=COLOR(RGB,. 8,.75,1),style=PATCHNOGRID):\np2 := plot(1/(1+x^2),x=0..5):\np3 := plo t([[4,0],[4,1/17]],color=black):\nplots[display]([p1,p2,p3],view=[0..5 ,0..1.1],labels=[x,y]);" }}{PARA 13 "" 1 "" {GLPLOT2D 394 202 202 {PLOTDATA 2 "6'-%)POLYGONSG6%7B7$$\"\"!F)F(7$F($\"\"\"F)7$$\"+J>7V9!#5 $\"+^'))fz*F07$$\"+2Yx)p#F0$\"+%)y5@$*F07$$\"+mt)36%F0$\"+3UOa&)F07$$ \"+C0OKbF0$\"+)pcll(F07$$\"+pw2ZpF0$\"+(p@[u'F07$$\"+f:qe#)F0$\"+'y&3X fF07$$\"+C@\"oh*F0$\"+h>E&>&F07$$\"+,s8-6!\"*$\"+n6F:XF07$$\"+YF9U7FT$ \"+C#*[KRF07$$\"+ec:'Q\"FT$\"+JT)HU$F07$$\"+4B+8:FT$\"+X^FSIF07$$\"+%f ,el\"FT$\"+QqfsEF07$$\"+Gs=*z\"FT$\"+#*)=,O#F07$$\"+aeOP>FT$\"+2Lw.@F0 7$$\"+%GXG1#FT$\"+Q)HG!>F07$$\"+jC07AFT$\"+^y(op\"F07$$\"+;0XQBFT$\"+A \"*)fa\"F07$$\"+*\\\"[&[#FT$\"+8)>KR\"F07$$\"+L)Gch#FT$\"+/REv7F07$$\" +'3@%eFFT$\"+e.fh6F07$$\"+xIR%*GFT$\"+:5Qm5F07$$\"+'eli.$FT$\"+O2y&y*! 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" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 39 "Int(1/(1+x^2),x=0..infinity);\nvalue(%); " }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%$IntG6$*&\"\"\"F',&F'F'*$)%\"xG \"\"#F'F'!\"\"/F+;\"\"!%)infinityG" }}{PARA 11 "" 1 "" {XPPMATH 20 "6# ,$%#PiG#\"\"\"\"\"#" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 9 "Example 4" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{PARA 0 " " 0 "" {TEXT -1 32 "Consider the improper integral: " }{XPPEDIT 18 0 " Int(1/x,x = 1 .. infinity);" "6#-%$IntG6$*&\"\"\"F'%\"xG!\"\"/F(;F'%)i nfinityG" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 141 "This integra l is illustrated by the following picture, if we imagine both the grap h and the shaded area extending indefinitely to the right. " }}{PARA 0 "" 0 "" {TEXT -1 19 "First we calculate " }{XPPEDIT 18 0 "Int(1/x,x \+ = 1 .. 4)" "6#-%$IntG6$*&\"\"\"F'%\"xG!\"\"/F(;F'\"\"%" }{TEXT -1 55 " , which is the area of the region bounded by the curve " }{XPPEDIT 18 0 "y = 1/x;" "6#/%\"yG*&\"\"\"F&%\"xG!\"\"" }{TEXT -1 6 ", the " } {TEXT 298 1 "x" }{TEXT -1 11 " axis, the " }{TEXT 299 1 "y" }{TEXT -1 19 " axis and the line " }{XPPEDIT 18 0 "y=4" "6#/%\"yG\"\"%" }{TEXT -1 3 ". " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 377 "pts:=[[1,0],op(op(1,op(1,plot(1/x,x=1..4,adaptive=fa lse,\n numpoints=30)))),[4,0]]:\np1 := plots[ polygonplot](pts,color=COLOR(RGB,.8,.75,1),\n \+ style=PATCHNOGRID):\np2 := plot([[[1,0],[1,1]],[[4,0],[4,1/4]]],style =LINE,color=black):\np3 := plot(1/x,x=0.5..5):\nplots[display]([p1,p2, p3],tickmarks=[4,3],view=[0..5,0..2],labels=[x,y]);" }}{PARA 13 "" 1 " " {GLPLOT2D 427 205 205 {PLOTDATA 2 "6)-%)POLYGONSG6%7B7$$\"\"\"\"\"!$ F*F*7$F(F(7$$\"+XTB36!\"*$\"+AROB!*!#57$$\"+'43C?\"F0$\"+RRk;$)F37$$\" +_lJ38F0$\"+Q,TVwF37$$\"+Rq#\\T\"F0$\"+EA]nqF37$$\"+D3.@:F0$\"+=()[ulF 37$$\"+;F0$\"+6m6vhF37$$\"+44E@F0$\"+qj.x^F37$$\"+VnhR?F0$\"+$*=)G!\\F37$$\"+K^=C#F0$\"+c$*fgWF37$$\"+@/R\\BF0$\"+YKUcUF37$$\"+!RCI X#F0$\"+9.gwSF37$$\"+jR8ZDF0$\"+z6)f#RF37$$\"+[$R!fEF0$\"+%oc2w$F37$$ \"+()y$Qv#F0$\"+feeKF37$$\"+3[zqJF0$\"+\"R$y`JF37$$\"+!>*>xKF0 $\"+$3'Q^IF37$$\"+G;\"\\P$F0$\"+[0/jHF37$$\"+$p2.[$F0$\"+n\"4L(GF37$$ \"+%e%y*e$F0$\"+uCo&y#F37$$\"+RX3&o$F0$\"+_>k8FF37$$\"+,6,)y$F0$\"+'y2 *REF37$$\"+7VM%*QF0$\"+mj#yc#F37$$\"\"%F*$\"+++++DF37$FftF+-%&COLORG6& %$RGBG$\"\")!\"\"$\"#v!\"#F)-%&STYLEG6#%,PATCHNOGRIDG-%'CURVESG6%7$F'F ,-%'COLOURG6&F^uF*F*F*-Ffu6#%%LINEG-Fju6%7$Fjt7$Fft$\"3++++++++D!#=F]v F`v-Fju6$7W7$$\"3++++++++]Fiv$\"\"#F*7$$\"3Q++vofV!\\&Fiv$\"3A7[%)))*[ 8#=!#<7$$\"3o****\\P>(3)fFiv$\"39XHb5q*>n\"Fgw7$$\"3%**\\i:l(f2kFiv$\" 3kpc*RIZ1c\"Fgw7$$\"3?+]ilLKMoFiv$\"3Yw\")GTE?j9Fgw7$$\"3C+v$41@UJ(Fiv $\"3/197&e*>n8Fgw7$$\"3E++Dc(=Tz(Fiv$\"3qWkL&o=IG\"Fgw7$$\"3i*****\\_( >x#)Fiv$\"3LAw$*y$Q\"37Fgw7$$\"31++v$Hw-w)Fiv$\"3AQ$[v];:9\"Fgw7$$\"3 \\**\\7`=%=s*Fiv$\"3$\\%Qkr;hG5Fgw7$$\"3++Dc@OLh5Fgw$\"3%HWwR*z5A%*Fiv 7$$\"3%**\\i!*pUO:\"Fgw$\"3)e#*ew5&>o')Fiv7$$\"3-+D1z)3\"\\7Fgw$\"3UaZ nUsq0!)Fiv7$$\"3#**\\7y*)oUM\"Fgw$\"3k&\\<:&y)*QuFiv7$$\"36++vtE:U9Fgw $\"3!4)e'[gyS$pFiv7$$\"3'***\\(ovo$G:Fgw$\"3:0H?%fBHa'Fiv7$$\"39++DYwU D;Fgw$\"3c)\\s64FA:'Fiv7$$\"3#****\\(o])Gs\"Fgw$\"3GEZb$pqIq=IFgw$\"3!GcZmNutD&Fiv7$ $\"3-++v3W].?Fgw$\"3STYP-VD\"*\\Fiv7$$\"3C+++&e:%*3#Fgw$\"3+([K_GFgy%F iv7$$\"3=+Dc12N*=#Fgw$\"3)Q8vPzFFiv7$$\"3y*\\iST\")fo$Fg w$\"3>6(fmm\")Hr#Fiv7$$\"3E++D;#RAy$Fgw$\"3xPSc;k$Rk#Fiv7$$\"3q*\\ilI5 G(QFgw$\"3iP#*QiU5#e#Fiv7$$\"3%)*\\7G>$[nRFgw$\"3`byKo&*[?DFiv7$$\"3/+ +vVK/gSFgw$\"3g'ziI&z-jCFiv7$$\"3M*\\i!R]%p:%Fgw$\"3NnH7`Fh0CFiv7$$\"3 ]+++&)HF]UFgw$\"3?ENh_+z_BFiv7$$\"3g**\\P*G9dM%Fgw$\"33qYz$o<6I#Fiv7$$ \"3E+Dc\"Hl.W%Fgw$\"3[^$H#op1_AFiv7$$\"3x****\\K(Rt_%Fgw$\"3j^+J]E!)3A Fiv7$$\"3p**\\(oDAqi%Fgw$\"35?)R8=<7;#Fiv7$$\"3W+++&\\zhr%Fgw$\"32au+ \\.O?@Fiv7$$\"3m*\\ilqR7\"[Fgw$\"3k " 0 "" {MPLTEXT 1 0 1 ";" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 256 "" 0 "" {TEXT -1 1 " " } {XPPEDIT 18 0 "Int(1/x,x=1..4)=ln(x)" "6#/-%$IntG6$*&\"\"\"F(%\"xG!\" \"/F);F(\"\"%-%#lnG6#F)" }{TEXT -1 1 " " }{XPPEDIT 18 0 "PIECEWISE([4, ``],[``, ``],[1, ``]);" "6#-%*PIECEWISEG6%7$\"\"%%!G7$F(F(7$\"\"\"F( " }{TEXT -1 1 " " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "` `=ln(4)-ln(1)" "6#/%!G,&-%#lnG6#\"\"%\"\"\"-F'6#F*!\"\"" }{TEXT -1 1 " " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "``=ln(4)" "6#/%! 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" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 26 "Int(1/x,x=1..4);\nvalue(%); " }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%$IntG6$*&\"\"\"F'%\"xG!\"\"/F(;F '\"\"%" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,$-%#lnG6#\"\"#F'" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{GLPLOT2D 356 233 233 {PLOTDATA 2 "6/-%)POLYGONSG6%7B7$$\"\"\"\"\"!$F*F*7$F(F(7$$\"+XTB36!\" *$\"+AROB!*!#57$$\"+'43C?\"F0$\"+RRk;$)F37$$\"+_lJ38F0$\"+Q,TVwF37$$\" +Rq#\\T\"F0$\"+EA]nqF37$$\"+D3.@:F0$\"+=()[ulF37$$\"+;F0$\"+6m6vhF 37$$\"+44E@F0$\"+qj.x ^F37$$\"+VnhR?F0$\"+$*=)G!\\F37$$\"+K^=C#F0$ \"+c$*fgWF37$$\"+@/R\\BF0$\"+YKUcUF37$$\"+!RCIX#F0$\"+9.gwSF37$$\"+jR8 ZDF0$\"+z6)f#RF37$$\"+[$R!fEF0$\"+%oc2w$F37$$\"+()y$Qv#F0$\"+f eeKF37$$\"+3[zqJF0$\"+\"R$y`JF37$$\"+!>*>xKF0$\"+$3'Q^IF37$$\"+G;\"\\P $F0$\"+[0/jHF37$$\"+$p2.[$F0$\"+n\"4L(GF37$$\"+%e%y*e$F0$\"+uCo&y#F37$ $\"+RX3&o$F0$\"+_>k8FF37$$\"+,6,)y$F0$\"+'y2*REF37$$\"+7VM%*QF0$\"+mj# yc#F37$$\"\"%F*$\"+++++DF37$FftF+-%&COLORG6&%$RGBG$\"\")!\"\"$\"#v!\"# F)-%&STYLEG6#%,PATCHNOGRIDG-%'CURVESG6%7$F'F,-%'COLOURG6&F^uF*F*F*-Ffu 6#%%LINEG-Fju6%7$Fjt7$Fft$\"3++++++++D!#=F]vF`v-Fju6%7$7$$!3!********* ******H!#>F(7$F+F(F]vF`v-Fju6$7V7$$\"3w**************fFiv$\"3ummmmmmm; !#<7$$\"3ammm\"RP&zkFiv$\"3(z\"=Va+KV:Fjw7$$\"3ILLL$yu!fpFiv$\"3+xZ*3j spV\"Fjw7$$\"3S***\\PqFjw$\"3+'Gm3BSP2&Fiv7$$\"3smmm*>$4q?Fjw$\"3)zZtpP+2 $[Fiv7$$\"3ummmQ_4a@Fjw$\"3bo\\lu-KUYFiv7$$\"3=++vz&4=D#Fjw$\"31buTQG( 3W%Fiv7$$\"3dmmmGLIQBFjw$\"3U*\\]`70mF%Fiv7$$\"34++vz1?LCFjw$\"3#>8g#) 38)4TFiv7$$\"3B++D*RlN_#Fjw$\"3O5+G_ukiRFiv7$$\"3,nm;/:&yh#Fjw$\"3Pt*y CPE*>QFiv7$$\"3'pm;9eNWq#Fjw$\"3/>!4\"4#Hwp$Fiv7$$\"3YLL$e?Gyz#Fjw$\"3 Ox]Nn3?uNFiv7$$\"3SLL3&*o$[*GFjw$\"3I/fgcgUaMFiv7$$\"3O++DWKGzHFjw$\"3 G[R#Fiv7$$\"3oLLL &eLpE%Fjw$\"3zw#pjQ.OM#Fiv7$$\"3'RLL=(RDgVFjw$\"35K9b(QWMH#Fiv7$$\"3An mT=;!GX%Fjw$\"3i\"RE77xdC#Fiv7$$\"3(******RHVy`%Fjw$\"3La!fX-!p.AFiv7$ $\"3&omm6l5`j%Fjw$\"3iP;36FNd@Fiv7$$\"3vLLLbelw?Fiv7$$\"3X++D[sR/\\Fjw$\"36IY*[a')*Q?F iv7$$\"\"&F*$\"35+++++++?FivF]v-%%TEXTG6$7$F($!\"(FduQ\"16\"-F[hl6$7$F ftF^hlQ\"RFahl-F[hl6$7$Ffgl$!\"&FduQ\"xFahl-F[hl6$7$$FauFau$\"$Z\"FduQ \"yFahl-F[hl6$7$F_ilF(F`hl-%*AXESTICKSG6$F*F*-%+AXESLABELSG6%%\"xG%\"y G-%%FONTG6#%(DEFAULTG-%%VIEWG6$;F_ilFfgl;F^hl$\"#:Fau" 1 2 0 1 10 0 2 9 1 4 2 1.000000 45.000000 45.000000 0 0 "Curve 1" "Curve 2" "Curve 3 " "Curve 4" "Curve 5" "Curve 6" "Curve 7" "Curve 8" "Curve 9" "Curve 1 0" }}{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 26 "The area under the \+ graph " }{XPPEDIT 18 0 "y = 1/x;" "6#/%\"yG*&\"\"\"F&%\"xG!\"\"" } {TEXT -1 6 " from " }{XPPEDIT 18 0 "x = 1;" "6#/%\"xG\"\"\"" }{TEXT -1 4 " to " }{XPPEDIT 18 0 "x = R" "6#/%\"xG%\"RG" }{TEXT -1 5 " is: \+ " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "Int(1/x,x = 1 .. \+ R) = ln(x);" "6#/-%$IntG6$*&\"\"\"F(%\"xG!\"\"/F);F(%\"RG-%#lnG6#F)" } {TEXT -1 1 " " }{XPPEDIT 18 0 "PIECEWISE([R, ``],[``, ``],[1, ``]);" " 6#-%*PIECEWISEG6%7$%\"RG%!G7$F(F(7$\"\"\"F(" }{TEXT -1 1 " " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "`` = ln(R)-ln(1)" "6#/%!G,& -%#lnG6#%\"RG\"\"\"-F'6#F*!\"\"" }{TEXT -1 1 " " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "``= ln(R)" "6#/%!G-%#lnG6#%\"RG" } {TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 19 "As the upper limit " }{TEXT 300 1 "R" }{TEXT -1 71 " of t he integral tends to infinity, the value of the integral tends to " } {XPPEDIT 18 0 "infinity;" "6#%)infinityG" }{TEXT -1 1 "." }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "Int(1/x,x = 1 .. infinity)=Limi t(Int(1/x,x = 1 .. R),R=infinity)" "6#/-%$IntG6$*&\"\"\"F(%\"xG!\"\"/F );F(%)infinityG-%&LimitG6$-F%6$*&F(F(F)F*/F);F(%\"RG/F6F-" }{TEXT -1 1 " " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "``=Limit(ln(R ),R=infinity)" "6#/%!G-%&LimitG6$-%#lnG6#%\"RG/F+%)infinityG" }{TEXT -1 1 " " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "``=infinit y" "6#/%!G%)infinityG" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 47 "In this case we say that the improper integral " }{XPPEDIT 18 0 "Int( 1/x,x=1..infinity)" "6#-%$IntG6$*&\"\"\"F'%\"xG!\"\"/F(;F'%)infinityG " }{TEXT -1 1 " " }{TEXT 260 20 "diverges to infinity" }{TEXT -1 2 ". \+ " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 60 "Int(1/x,x=1..R);\nvalue( %);\nInt(1/x,x=1..infinity);\nvalue(%);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%$IntG6$*&\"\"\"F'%\"xG!\"\"/F(;F'%\"RG" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%#lnG6#%\"RG" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%$Int G6$*&\"\"\"F'%\"xG!\"\"/F(;F'%)infinityG" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%)infinityG" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 " > " 0 "" {MPLTEXT 1 0 33 "Int(1/x,x=1..infinity);\nvalue(%);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%$IntG6$*&\"\"\"F'%\"xG!\"\"/F(;F'%)infinit yG" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%)infinityG" }}}{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 28 "A special improper integral " }}{EXCHG {PARA 0 "> " 0 " " {MPLTEXT 1 0 1 ";" }}}{PARA 0 "" 0 "" {TEXT -1 96 "The following res ult will be useful in the determination of formulas for the Laplace tr ansform. " }}{PARA 256 "" 0 "" {XPPEDIT 18 0 "Int(x^n/exp(a*x),x = 0 . . infinity) = n!/(a^(n+1));" "6#/-%$IntG6$*&)%\"xG%\"nG\"\"\"-%$expG6# *&%\"aGF+F)F+!\"\"/F);\"\"!%)infinityG*&-%*factorialG6#F*F+)F0,&F*F+F+ F+F1" }{TEXT -1 15 " ------- (i), " }}{PARA 256 "" 0 "" {TEXT -1 1 " \+ " }{TEXT 275 13 "_____________" }{TEXT -1 14 " " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 6 "where " }{TEXT 271 1 "n" }{TEXT -1 27 " is a positive integer and " }{TEXT 272 1 "a" }{TEXT -1 30 " is a positive real constant. " }}{PARA 0 "" 0 "" {TEXT -1 61 "In order to establish the result (i) we need to make use of: " }}{PARA 256 "" 0 "" {TEXT -1 2 " " }{XPPEDIT 18 0 "Limit(x^n/exp(a*x) ,x = infinity) = 0;" "6#/-%&LimitG6$*&)%\"xG%\"nG\"\"\"-%$expG6#*&%\"a GF+F)F+!\"\"/F)%)infinityG\"\"!" }{TEXT -1 15 " ------- (ii)," }} {PARA 256 "" 0 "" {TEXT -1 1 " " }{TEXT 276 10 "__________" }{TEXT -1 15 " " }}{PARA 0 "" 0 "" {TEXT -1 6 "where " }{TEXT 273 1 "n" }{TEXT -1 27 " is a positive integer and " }{TEXT 274 1 "a" } {TEXT -1 30 " is a positive real constant. " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 19 "Discussion of (ii) " }}{PARA 0 "" 0 "" {TEXT -1 28 "Roughly speaking the result " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "Limit(x^n/exp(a*x),x = infinity) = 0" "6#/-%& LimitG6$*&)%\"xG%\"nG\"\"\"-%$expG6#*&%\"aGF+F)F+!\"\"/F)%)infinityG\" \"!" }{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 33 " follows because t he exponential " }{XPPEDIT 18 0 "exp(a*x)" "6#-%$expG6#*&%\"aG\"\"\"% \"xGF(" }{TEXT -1 23 " in the denominator of " }{XPPEDIT 18 0 "x^n/exp (a*x)" "6#*&)%\"xG%\"nG\"\"\"-%$expG6#*&%\"aGF'F%F'!\"\"" }{TEXT -1 53 " will eventually grow much larger than the numerator " }{XPPEDIT 18 0 "x^n" "6#)%\"xG%\"nG" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 23 "As an example consider " }{XPPEDIT 18 0 "Limit(x^5*exp(-x/8),x \+ = infinity);" "6#-%&LimitG6$*&%\"xG\"\"&-%$expG6#,$*&F'\"\"\"\"\")!\" \"F0F./F'%)infinityG" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 36 " A graph may initially suggest that " }{XPPEDIT 18 0 "f(x) = x^5*exp(- x/8);" "6#/-%\"fG6#%\"xG*&F'\"\"&-%$expG6#,$*&F'\"\"\"\"\")!\"\"F1F/" }{TEXT -1 38 " tends to infinity as increases . . . 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[hY'FD$\"3xG$pgjWt.&Fhr7$$\"3iKLL$Qx$omFD$\"3%)[XmSaAHdFhr7$$\"3Y+++v. I%)oFD$\"3<<(4In=*RlFhr7$$\"3?mm\"zpe*zqFD$\"3#p%f3#fX=M(Fhr7$$\"3;,++ D\\'QH(FD$\"3k73%=B?_H)Fhr7$$\"3%HL$e9S8&\\(FD$\"3wr(*pnO^o#*Fhr7$$\"3 s++D1#=bq(FD$\"3]@%[![\\\"o.\"!#87$$\"3\"HLL$3s?6zFD$\"3'Gyn^%=w_6Fdw7 $$\"3a***\\7`Wl7)FD$\"3_(3?O**4MG\"Fdw7$$\"3enmmm*RRL)FD$\"3**o[V/$)\\ =9Fdw7$$\"3%zmmTvJga)FD$\"3]S[f-)Qjc\"Fdw7$$\"3]MLe9tOc()FD$\"3Kp.j()z \"Hs\"Fdw7$$\"31,++]Qk\\*)FD$\"3/uICXnxv=Fdw7$$\"3![LL3dg6<*FD$\"3%y\" 4k2!4=1#Fdw7$$\"3%ymmmw(Gp$*FD$\"31iGukqAQAFdw7$$\"3C++D\"oK0e*FD$\"3Y &HAlkypV#Fdw7$$\"35,+v=5s#y*FD$\"3g;40h8qPEFdw7$$\"#5F)$\"3Y,>goz/lGFd w-%'COLOURG6&%$RGBG$Fiz!\"\"F(F(-%+AXESLABELSG6$Q\"x6\"Q!Ff[l-%%VIEWG6 $;F(Fhz%(DEFAULTG" 1 2 0 1 10 0 2 9 1 4 2 1.000000 45.000000 45.000000 0 0 "Curve 1" }}}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 " " 0 "" {TEXT -1 33 "However, taking larger values of " }{TEXT 277 1 "x " }{TEXT -1 51 " shows that a maximum value is reached after which " } {XPPEDIT 18 0 "f(x)" "6#-%\"fG6#%\"xG" }{TEXT -1 24 " appears to appro ach 0. " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 29 "plot(x^5*exp(-x/8),x=0..120);" }}{PARA 13 "" 1 "" {GLPLOT2D 400 300 300 {PLOTDATA 2 "6%-%'CURVESG6$7_o7$$\"\"!F)F(7$$\"3 *)*******\\ech#!#<$\"3'e_\"pV!e*G))!#;7$$\"39++]PPf`PF-$\"3KbthRh&3m%! #:7$$\"3%******\\(*G:*[F-$\"3u\"y)yG$>%>:!#97$$\"3u****\\ihDrhF-$\"38' )*))pjn&QTF<7$$\"3c******\\L)4X(F-$\"3%G()Q69E&[!*F<7$$\"3M,+++M>R()F- $\"3i]fq[Zv4cXY^+d6!#77$$\"3#***** \\F%o*>;F0$\"3IT5hXOms9Ffn7$$\"3(*****\\(>ZIu\"F0$\"39y,&H`i4#=Ffn7$$ \"3+++]PaLq=F0$\"3'4q\\Pi'G4AFfn7$$\"3-++]xOi(*>F0$\"3zc3:l)>*=EFfn7$$ \"3?++]-P]C@F0$\"3o))GbT,iSIFfn7$$\"3-++]FPQ^AF0$\"3I$GHGW8xY$Ffn7$$\" 3))***\\(Ga*=Q#F0$\"3'fTQ?&oV/RFfn7$$\"32+++IrS7DF0$\"3&4(*e'RLcIVFfn7 $$\"3#*****\\2>OFEF0$\"3U`tO/%\\7p%Ffn7$$\"37+++&o;Bu#F0$\"3gaS1I(>N.& Ffn7$$\"3I++]P&G<(GF0$\"3'Q.SsTl>R&Ffn7$$\"37+++!RS6+$F0$\"3m6<\"yGEvr &Ffn7$$\"3s******>O3JJF0$\"3EDOF0$\"3#)y2 hTt!*RnFfn7$$\"30++]xq!*QPF0$\"3^*oEBMHK#oFfn7$$\"30+]i!e;l!QF0$\"3-Ro omW2eoFfn7$$\"31++v$3ET(QF0$\"3?p4p?:B#)oFfn7$$\"31+](oeN<%RF0$\"3e52C 5I'f*oFfn7$$\"31+++!4X$4SF0$\"3a`tLDPc**oFfn7$$\"3!)****\\2#>m1%F0$\"3 q`cf4s#\\*oFfn7$$\"3B+++DL*Q7%F0$\"3_#G5WziM)oFfn7$$\"3)*****\\Uu;\"=% F0$\"3qQ(=()R#RloFfn7$$\"3q******f:WQUF0$\"3c^'>[qZ4%oFfn7$$\"3k***\\( )Q)orVF0$\"3e!Qo3ak2w'Ffn7$$\"3d****\\<_$\\]%F0$\"3Sv<1yJa]mFfn7$$\"3m ******fs#3u%F0$\"3s5%R[q_=R'Ffn7$$\"33++]<#Q'**\\F0$\"3=92&*\\Z@LgFfn7 $$\"39++]_u3Y_F0$\"3'\\0\"e/`aRcFfn7$$\"3,+++v8B.bF0$\"3Sb[S?ap%>&Ffn7 $$\"3d++]n(p$RdF0$\"33!)yLX$**3x%Ffn7$$\"3C+++Dp2%*fF0$\"3C9%G3x+9J%Ff n7$$\"3#*****\\xgkeiF0$\"3VA5n9N*R%QFfn7$$\"3[****\\-V&*)['F0$\"3!*yVA KSC`MFfn7$$\"3h+++&\\$pPnF0$\"3a-'eYjeS0$Ffn7$$\"3w******>am%*pF0$\"3) *3jt)ex3n#Ffn7$$\"3O+++:B1YsF0$\"3G_!)4R*4tK#Ffn7$$\"3'*****\\PQ&3` !)fkFL7$$\"39******\\'[M\\*F0$\"3rx**Q\\Y*GT&FL7$$\"3;++]PM&=v*F0$\"3I B;Bqg$>[%FL7$$\"3/+++'zs++\"F6$\"3Tu(Hs.>Ys$FL7$$\"33++]5Q_D5F6$\"3)y8 F?/mC2$FL7$$\"33++vxSw]5F6$\"3L&Q())*fI3`#FL7$$\"37+++is&R2\"F6$\"3;)) \\C\\)QB6#FL7$$\"36++]o#R05\"F6$\"3w0gE?l=7`9V7\"F6$ \"3p`)f2x[aT\"FL7$$\"35++v@Rm\\6F6$\"3mx\"oIgXE:\"FL7$$\"31++DAl#R<\"F 6$\"3!4'4*o8FyW*F<7$$\"$?\"F)$\"3\"H.6:iG=h(F<-%'COLOURG6&%$RGBG$\"#5! \"\"F(F(-%+AXESLABELSG6$Q\"x6\"Q!F_al-%%VIEWG6$;F(F``l%(DEFAULTG" 1 2 0 1 10 0 2 9 1 4 2 1.000000 45.000000 45.000000 0 0 "Curve 1" }}}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 27 "In fact the derivative of " } {XPPEDIT 18 0 "f(x)=x^5*exp(-x/8)" "6#/-%\"fG6#%\"xG*&F'\"\"&-%$expG6# ,$*&F'\"\"\"\"\")!\"\"F1F/" }{TEXT -1 17 " with respect to " }{TEXT 279 1 "x" }{TEXT -1 5 " is: " }}{PARA 256 "" 0 "" {TEXT -1 4 "f '(" } {TEXT 278 1 "x" }{TEXT -1 4 ") = " }{XPPEDIT 18 0 "5*x^4*exp(-x/8)-x^5 /8;" "6#,&*(\"\"&\"\"\"*$%\"xG\"\"%F&-%$expG6#,$*&F(F&\"\")!\"\"F0F&F& *&F(F%F/F0F0" }{TEXT -1 1 " " }{XPPEDIT 18 0 "exp(-x/8) = ``(40-x);" " 6#/-%$expG6#,$*&%\"xG\"\"\"\"\")!\"\"F,-%!G6#,&\"#SF*F)F," }{TEXT -1 1 " " }{XPPEDIT 18 0 "x^4/8" "6#*&%\"xG\"\"%\"\")!\"\"" }{TEXT -1 1 " \+ " }{XPPEDIT 18 0 "exp(-x/8)" "6#-%$expG6#,$*&%\"xG\"\"\"\"\")!\"\"F+" }{TEXT -1 3 " . " }}{PARA 0 "" 0 "" {TEXT -1 55 "The sign of the deriv ative is determined by the sign of" }{XPPEDIT 18 0 "``(40-x)" "6#-%!G6 #,&\"#S\"\"\"%\"xG!\"\"" }{TEXT -1 29 ", since the remaining factor " }{XPPEDIT 18 0 "x^4/8" "6#*&%\"xG\"\"%\"\")!\"\"" }{TEXT -1 1 " " } {XPPEDIT 18 0 "exp(-x/8)" "6#-%$expG6#,$*&%\"xG\"\"\"\"\")!\"\"F+" } {TEXT -1 25 " is always positive when " }{TEXT 280 1 "x" }{TEXT -1 14 " is positive. " }}{PARA 0 "" 0 "" {TEXT -1 9 "Thus f '(" }{TEXT 281 1 "x" }{TEXT -1 19 ") is positive when " }{XPPEDIT 18 0 "040" "6#2\"#S%\"xG" }{TEXT -1 2 ". " }} {PARA 0 "" 0 "" {TEXT -1 81 "This is consistent with what the last gra ph shows, and so supports the idea that " }{XPPEDIT 18 0 "Limit(x^5*ex p(-8*x),x = infinity)=0" "6#/-%&LimitG6$*&%\"xG\"\"&-%$expG6#,$*&\"\") \"\"\"F(F0!\"\"F0/F(%)infinityG\"\"!" }{TEXT -1 52 ". However it does \+ not demonstrate conclusively that " }{XPPEDIT 18 0 "f(x) = x^5*exp(-x/ 8);" "6#/-%\"fG6#%\"xG*&F'\"\"&-%$expG6#,$*&F'\"\"\"\"\")!\"\"F1F/" } {TEXT -1 16 " tends to 0, as " }{TEXT 287 1 "x" }{TEXT -1 40 " tends t o infinity, but only shows that " }{XPPEDIT 18 0 "f(x)" "6#-%\"fG6#%\" xG" }{TEXT -1 27 " is eventually decreasing. " }}{PARA 0 "" 0 "" {TEXT -1 71 "A similar conclusion can be reached for a general functio n of the form " }{XPPEDIT 18 0 "f(x)=x^n/exp(a*x)" "6#/-%\"fG6#%\"xG*& )F'%\"nG\"\"\"-%$expG6#*&%\"aGF+F'F+!\"\"" }{TEXT -1 8 ", where " } {TEXT 282 1 "n" }{TEXT -1 27 " is a positive integer and " }{TEXT 283 1 "a" }{TEXT -1 30 " is a positive real constant. " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 49 "The usual way to obtain a proof of the fact that " }{XPPEDIT 18 0 "Limit(x^n/exp(a*x),x = infin ity)=0" "6#/-%&LimitG6$*&)%\"xG%\"nG\"\"\"-%$expG6#*&%\"aGF+F)F+!\"\"/ F)%)infinityG\"\"!" }{TEXT -1 20 " is to make use of " }{TEXT 260 17 "L'Hospital's rule" }{TEXT -1 23 " in the following form:" }}{PARA 256 "" 0 "" {TEXT -1 3 "If " }{XPPEDIT 18 0 "u(x)->infinity" "6#f*6#-% \"uG6#%\"xG7\"6$%)operatorG%&arrowG6\"%)infinityGF-F-F-" }{TEXT -1 5 " and " }{XPPEDIT 18 0 "v(x)->infinity" "6#f*6#-%\"vG6#%\"xG7\"6$%)oper atorG%&arrowG6\"%)infinityGF-F-F-" }{TEXT -1 5 " as " }{XPPEDIT 18 0 "x->infinity" "6#f*6#%\"xG7\"6$%)operatorG%&arrowG6\"%)infinityGF*F*F* " }{TEXT -1 8 ", then " }{XPPEDIT 18 0 "Limit(u(x)/v(x),x=infinity)=L imit(u*`'`(x)/(v*`'`(x)),x=infinity)" "6#/-%&LimitG6$*&-%\"uG6#%\"xG\" \"\"-%\"vG6#F+!\"\"/F+%)infinityG-F%6$*(F)F,-%\"'G6#F+F,*&F.F,-F76#F+F ,F0/F+F2" }{TEXT -1 2 ". " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{TEXT 286 39 "_______________________________________" }{TEXT -1 1 " " }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 11 "Given tha t " }{TEXT 284 1 "a" }{TEXT -1 58 " is positive we can apply this rule repeatedly to obtain: " }}{PARA 256 "" 0 "" {TEXT -1 2 " " } {XPPEDIT 18 0 "Limit(x^n/exp(a*x),x = infinity) = Limit(n*x^(n-1)/(a*e xp(a*x)),x = infinity)" "6#/-%&LimitG6$*&)%\"xG%\"nG\"\"\"-%$expG6#*&% \"aGF+F)F+!\"\"/F)%)infinityG-F%6$*(F*F+)F),&F*F+F+F1F+*&F0F+-F-6#*&F0 F+F)F+F+F1/F)F3" }{TEXT -1 1 " " }{XPPEDIT 18 0 "``= n/a" "6#/%!G*&%\" nG\"\"\"%\"aG!\"\"" }{TEXT -1 1 " " }{XPPEDIT 18 0 "Limit(x^(n-1)/exp( a*x),x = infinity)" "6#-%&LimitG6$*&)%\"xG,&%\"nG\"\"\"F+!\"\"F+-%$exp G6#*&%\"aGF+F(F+F,/F(%)infinityG" }{TEXT -1 1 " " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "``= n*(n-1)/a^2" "6#/%!G*(%\"nG\"\"\",& F&F'F'!\"\"F'*$%\"aG\"\"#F)" }{TEXT -1 1 " " }{XPPEDIT 18 0 "Limit(x^( n-2)/exp(a*x),x = infinity)" "6#-%&LimitG6$*&)%\"xG,&%\"nG\"\"\"\"\"#! \"\"F+-%$expG6#*&%\"aGF+F(F+F-/F(%)infinityG" }{TEXT -1 1 " " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "``= n*(n-1)*(n-2)/a^3" "6#/ %!G**%\"nG\"\"\",&F&F'F'!\"\"F',&F&F'\"\"#F)F'*$%\"aG\"\"$F)" }{TEXT -1 1 " " }{XPPEDIT 18 0 "Limit(x^(n-3)/exp(a*x),x = infinity)" "6#-%&L imitG6$*&)%\"xG,&%\"nG\"\"\"\"\"$!\"\"F+-%$expG6#*&%\"aGF+F(F+F-/F(%)i nfinityG" }{TEXT -1 1 " " }}{PARA 256 "" 0 "" {TEXT -1 10 " . . . \+ " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "``= n!/a^n" "6#/% !G*&-%*factorialG6#%\"nG\"\"\")%\"aGF)!\"\"" }{TEXT -1 1 " " } {XPPEDIT 18 0 "Limit(1/exp(a*x),x = infinity)" "6#-%&LimitG6$*&\"\"\"F '-%$expG6#*&%\"aGF'%\"xGF'!\"\"/F-%)infinityG" }{TEXT -1 1 " " }} {PARA 0 "" 0 "" {TEXT -1 61 "Since this last limit is zero, we obtain \+ the desired result. " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 " " {TEXT 285 9 "Examples:" }{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 1 " " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 42 "Limit(x^5*exp(-x/8), x=infinity);\nvalue(%);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%&LimitG6$ *&)%\"xG\"\"&\"\"\"-%$expG6#,$F(#!\"\"\"\")F*/F(%)infinityG" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#\"\"!" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 45 "Limit(x^21*exp(-x/37),x=infi nity);\nvalue(%);\n" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%&LimitG6$*&)% \"xG\"#@\"\"\"-%$expG6#,$F(#!\"\"\"#PF*/F(%)infinityG" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#\"\"!" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 18 "Discussion of (i) " }}{PARA 0 "" 0 "" {TEXT -1 77 "As an \+ illustration of the ideas involved we start with the specific example: " }{XPPEDIT 18 0 "Int(x*exp(-x/3),x=0..infinity)" "6#-%$IntG6$*&%\"xG \"\"\"-%$expG6#,$*&F'F(\"\"$!\"\"F/F(/F';\"\"!%)infinityG" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 39 "Using the integration by parts f ormula " }{XPPEDIT 18 0 "Int(u*``(dv/dx),x)=u*v-Int(v*``(du/dx),x)" "6 #/-%$IntG6$*&%\"uG\"\"\"-%!G6#*&%#dvGF)%#dxG!\"\"F)%\"xG,&*&F(F)%\"vGF )F)-F%6$*&F4F)-F+6#*&%#duGF)F/F0F)F1F0" }{TEXT -1 8 " gives: " }} {PARA 258 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "Int(x*exp(-x/3),x)" " 6#-%$IntG6$*&%\"xG\"\"\"-%$expG6#,$*&F'F(\"\"$!\"\"F/F(F'" }{TEXT -1 9 " ... " }{XPPEDIT 18 0 "PIECEWISE([u=x,v=-3*exp(-x/3)],[du/dx=1, dv/dx=exp(-x/3)])" "6#-%*PIECEWISEG6$7$/%\"uG%\"xG/%\"vG,$*&\"\"$\"\" \"-%$expG6#,$*&F)F/F.!\"\"F5F/F57$/*&%#duGF/%#dxGF5F//*&%#dvGF/F:F5-F1 6#,$*&F)F/F.F5F5" }{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {PARA 256 "" 0 "" {TEXT -1 2 " " }{XPPEDIT 18 0 "`` = Int(u*``(dv/dx) ,x);" "6#/%!G-%$IntG6$*&%\"uG\"\"\"-F$6#*&%#dvGF*%#dxG!\"\"F*%\"xG" } {XPPEDIT 18 0 " ``= u*v-Int(v*``(du/dx),x)" "6#/%!G,&*&%\"uG\"\"\"%\"v GF(F(-%$IntG6$*&F)F(-F$6#*&%#duGF(%#dxG!\"\"F(%\"xGF3" }{TEXT -1 1 " \+ " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 256 "" 0 "" {TEXT -1 1 " " } {XPPEDIT 18 0 "`` = -3*x*exp(-x/3)-Int(-3*exp(-x/3),x);" "6#/%!G,&*(\" \"$\"\"\"%\"xGF(-%$expG6#,$*&F)F(F'!\"\"F/F(F/-%$IntG6$,$*&F'F(-F+6#,$ *&F)F(F'F/F/F(F/F)F/" }{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "``=-3*x*exp(-x/3)-9 *exp(-x/3)" "6#/%!G,&*(\"\"$\"\"\"%\"xGF(-%$expG6#,$*&F)F(F'!\"\"F/F(F /*&\"\"*F(-F+6#,$*&F)F(F'F/F/F(F/" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 6 "Hence " }}{PARA 256 "" 0 " " {TEXT -1 1 " " }{XPPEDIT 18 0 "Int(x*exp(-x/3),x = 0 .. infinity)=Li mit(Int(x*exp(-x/3),x = 0 .. R),R=infinity)" "6#/-%$IntG6$*&%\"xG\"\" \"-%$expG6#,$*&F(F)\"\"$!\"\"F0F)/F(;\"\"!%)infinityG-%&LimitG6$-F%6$* &F(F)-F+6#,$*&F(F)F/F0F0F)/F(;F3%\"RG/FAF4" }{TEXT -1 1 " " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "``= 3" "6#/%!G\"\"$" } {TEXT -1 1 " " }{XPPEDIT 18 0 "Limit(R*exp(-R/3),R = infinity)" "6#-%& LimitG6$*&%\"RG\"\"\"-%$expG6#,$*&F'F(\"\"$!\"\"F/F(/F'%)infinityG" } {XPPEDIT 18 0 " ``- 9" "6#,&%!G\"\"\"\"\"*!\"\"" }{TEXT -1 1 " " } {XPPEDIT 18 0 "Limit(``(exp(-R/3)-1),R = infinity)" "6#-%&LimitG6$-%!G 6#,&-%$expG6#,$*&%\"RG\"\"\"\"\"$!\"\"F2F0F0F2/F/%)infinityG" }{TEXT -1 2 " " }{XPPEDIT 18 0 "``=9" "6#/%!G\"\"*" }{TEXT -1 2 ". " }} {PARA 0 "" 0 "" {TEXT -1 56 "Note that the first limit is 0 as a conse quence of (ii)." }}{PARA 0 "" 0 "" {TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 86 "student[intp arts](Int(x*exp(-x/3),x=0..R),x);\nvalue(%);\nLimit(%,R=infinity);\nva lue(%);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,&*&%\"RG\"\"\"-%$expG6#,$F %#!\"\"\"\"$F&!\"$-%$IntG6$,$-F(6#,$%\"xGF+F./F6;\"\"!F%F," }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,(*&%\"RG\"\"\"-%$expG6#,$F%#!\"\"\"\"$F&!\" $*&\"\"*F&F'F&F,F0F&" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%&LimitG6$,(* &%\"RG\"\"\"-%$expG6#,$F(#!\"\"\"\"$F)!\"$*&\"\"*F)F*F)F/F3F)/F(%)infi nityG" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#\"\"*" }}}{EXCHG {PARA 0 "> \+ " 0 "" {MPLTEXT 1 0 1 ";" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 " " 0 "" {TEXT -1 56 "Now consider the general case of the improper inte gral " }{XPPEDIT 18 0 "Int(x^n*exp(-a*x),x = 0 .. infinity)" "6#-%$In tG6$*&)%\"xG%\"nG\"\"\"-%$expG6#,$*&%\"aGF*F(F*!\"\"F*/F(;\"\"!%)infin ityG" }{TEXT -1 8 ", where " }{TEXT 288 1 "n" }{TEXT -1 27 " is a posi tive integer and " }{TEXT 289 1 "a" }{TEXT -1 29 " is a positive real \+ constant." }}{PARA 0 "" 0 "" {TEXT -1 44 "Using the integration by par ts formula with " }{XPPEDIT 18 0 "u=x^n" "6#/%\"uG)%\"xG%\"nG" }{TEXT -1 5 " and " }{XPPEDIT 18 0 "v=-exp(-a*x)/a" "6#/%\"vG,$*&-%$expG6#,$* &%\"aG\"\"\"%\"xGF-!\"\"F-F,F/F/" }{TEXT -1 9 " so that " }{XPPEDIT 18 0 "du/dx=n*x^(n-1)" "6#/*&%#duG\"\"\"%#dxG!\"\"*&%\"nGF&)%\"xG,&F*F &F&F(F&" }{TEXT -1 5 " and " }{XPPEDIT 18 0 "dv/dx=exp(-a*x)" "6#/*&%# dvG\"\"\"%#dxG!\"\"-%$expG6#,$*&%\"aGF&%\"xGF&F(" }{TEXT -1 10 ", we h ave " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "Int(x^n*exp(- a*x),x = 0 .. R);" "6#-%$IntG6$*&)%\"xG%\"nG\"\"\"-%$expG6#,$*&%\"aGF* F(F*!\"\"F*/F(;\"\"!%\"RG" }{TEXT -1 1 " " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "``= Int(u*``(dv/dx),x=0..R)" "6#/%!G-%$IntG6$ *&%\"uG\"\"\"-F$6#*&%#dvGF*%#dxG!\"\"F*/%\"xG;\"\"!%\"RG" }{XPPEDIT 18 0 " ``= u*v" "6#/%!G*&%\"uG\"\"\"%\"vGF'" }{TEXT -1 1 " " } {XPPEDIT 18 0 "PIECEWISE([R, ``],[``, ``],[0, ``]);" "6#-%*PIECEWISEG6 %7$%\"RG%!G7$F(F(7$\"\"!F(" }{XPPEDIT 18 0 " ``-Int(v*``(du/dx),x=0..R )" "6#,&%!G\"\"\"-%$IntG6$*&%\"vGF%-F$6#*&%#duGF%%#dxG!\"\"F%/%\"xG;\" \"!%\"RGF0" }{TEXT -1 1 " " }}{PARA 256 "" 0 "" {TEXT -1 1 " " } {XPPEDIT 18 0 "``=-x^n/a*exp(-a*x)" "6#/%!G,$*()%\"xG%\"nG\"\"\"%\"aG! \"\"-%$expG6#,$*&F+F*F(F*F,F*F," }{TEXT -1 1 " " }{XPPEDIT 18 0 "PIECE WISE([R, ``],[``, ``],[0, ``]);" "6#-%*PIECEWISEG6%7$%\"RG%!G7$F(F(7$ \"\"!F(" }{TEXT -1 1 " " }{XPPEDIT 18 0 "``-Int(-n*x^(n-1)/a*exp(-a*x) ,x = 0 .. R);" "6#,&%!G\"\"\"-%$IntG6$,$**%\"nGF%)%\"xG,&F+F%F%!\"\"F% %\"aGF/-%$expG6#,$*&F0F%F-F%F/F%F//F-;\"\"!%\"RGF/" }{TEXT -1 1 " " }} {PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "`` = -R^n/a*exp(-a*R) +n/a" "6#/%!G,&*()%\"RG%\"nG\"\"\"%\"aG!\"\"-%$expG6#,$*&F+F*F(F*F,F*F ,*&F)F*F+F,F*" }{TEXT -1 1 " " }{XPPEDIT 18 0 "Int(x^(n-1)*exp(-a*x),x = 0 .. R)" "6#-%$IntG6$*&)%\"xG,&%\"nG\"\"\"F+!\"\"F+-%$expG6#,$*&%\" aGF+F(F+F,F+/F(;\"\"!%\"RG" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 18 " Taking limits as " }{XPPEDIT 18 0 "R->infinity" "6#f*6#%\"RG7 \"6$%)operatorG%&arrowG6\"%)infinityGF*F*F*" }{TEXT -1 31 " and making use of (ii) gives: " }}{PARA 256 "" 0 "" {TEXT -1 2 " " }{XPPEDIT 18 0 "Int(x^n*exp(-a*x),x = 0 .. infinity) = n/a" "6#/-%$IntG6$*&)%\"x G%\"nG\"\"\"-%$expG6#,$*&%\"aGF+F)F+!\"\"F+/F);\"\"!%)infinityG*&F*F+F 1F2" }{TEXT -1 1 " " }{XPPEDIT 18 0 "Int(x^(n-1)*exp(-a*x),x = 0 .. in finity);" "6#-%$IntG6$*&)%\"xG,&%\"nG\"\"\"F+!\"\"F+-%$expG6#,$*&%\"aG F+F(F+F,F+/F(;\"\"!%)infinityG" }{TEXT -1 16 " ------- (iii). " }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 36 "Repeating the previous argument for " }{XPPEDIT 18 0 "Int(x^(n-1)*exp(-a*x),x = 0 .. infinity)" "6#-%$IntG6$*&)%\"xG,&%\"nG\"\"\"F+!\"\"F+-%$expG6#,$ *&%\"aGF+F(F+F,F+/F(;\"\"!%)infinityG" }{TEXT -1 13 " in place of " } {XPPEDIT 18 0 "Int(x^n*exp(-a*x),x = 0 .. infinity)" "6#-%$IntG6$*&)% \"xG%\"nG\"\"\"-%$expG6#,$*&%\"aGF*F(F*!\"\"F*/F(;\"\"!%)infinityG" } {TEXT -1 8 " gives: " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "Int(x^(n-1)*exp(-a*x),x = 0 .. infinity) = (n-1)/a" "6#/-%$IntG6$*& )%\"xG,&%\"nG\"\"\"F,!\"\"F,-%$expG6#,$*&%\"aGF,F)F,F-F,/F);\"\"!%)inf inityG*&,&F+F,F,F-F,F3F-" }{TEXT -1 1 " " }{XPPEDIT 18 0 "Int(x^(n-2)* exp(-a*x),x = 0 .. infinity)" "6#-%$IntG6$*&)%\"xG,&%\"nG\"\"\"\"\"#! \"\"F+-%$expG6#,$*&%\"aGF+F(F+F-F+/F(;\"\"!%)infinityG" }{TEXT -1 15 " ------- (iv). " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 70 "Substituting for the integral on the right of (iii) using (iv) gives: " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "Int( x^n*exp(-a*x),x = 0 .. infinity) = n*(n-1)/(a^2);" "6#/-%$IntG6$*&)%\" xG%\"nG\"\"\"-%$expG6#,$*&%\"aGF+F)F+!\"\"F+/F);\"\"!%)infinityG*(F*F+ ,&F*F+F+F2F+*$F1\"\"#F2" }{TEXT -1 1 " " }{XPPEDIT 18 0 "Int(x^(n-2)*e xp(-a*x),x = 0 .. infinity);" "6#-%$IntG6$*&)%\"xG,&%\"nG\"\"\"\"\"#! \"\"F+-%$expG6#,$*&%\"aGF+F(F+F-F+/F(;\"\"!%)infinityG" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 30 "Continuing in this way gives: " }} {PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "Int(x^n*exp(-a*x),x = 0 .. infinity) = n!/(a^n)" "6#/-%$IntG6$*&)%\"xG%\"nG\"\"\"-%$expG6#, $*&%\"aGF+F)F+!\"\"F+/F);\"\"!%)infinityG*&-%*factorialG6#F*F+)F1F*F2 " }{TEXT -1 1 " " }{XPPEDIT 18 0 "Int(exp(-a*x),x = 0 .. infinity)" "6 #-%$IntG6$-%$expG6#,$*&%\"aG\"\"\"%\"xGF,!\"\"/F-;\"\"!%)infinityG" } {TEXT -1 14 " ------- (v). " }}{PARA 0 "" 0 "" {TEXT -1 5 " Now " }} {PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "Int(exp(-a*x),x = 0 . . infinity) = Limit(Int(exp(-a*x),x = 0 .. R),R=infinity)" "6#/-%$IntG 6$-%$expG6#,$*&%\"aG\"\"\"%\"xGF-!\"\"/F.;\"\"!%)infinityG-%&LimitG6$- F%6$-F(6#,$*&F,F-F.F-F//F.;F2%\"RG/F?F3" }{TEXT -1 1 " " }}{PARA 256 " " 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "``=Limit(``,R=infinity)" "6#/%!G -%&LimitG6$F$/%\"RG%)infinityG" }{XPPEDIT 18 0 "-exp(-a*x)/a" "6#,$*&- %$expG6#,$*&%\"aG\"\"\"%\"xGF+!\"\"F+F*F-F-" }{TEXT -1 1 " " } {XPPEDIT 18 0 "PIECEWISE([R,``],[0,``])" "6#-%*PIECEWISEG6$7$%\"RG%!G7 $\"\"!F(" }{TEXT -1 1 " " }}{PARA 256 "" 0 "" {TEXT -1 1 " " } {XPPEDIT 18 0 "`` = Limit(``(1/a-exp(-R*a)/a),R = infinity);" "6#/%!G- %&LimitG6$-F$6#,&*&\"\"\"F,%\"aG!\"\"F,*&-%$expG6#,$*&%\"RGF,F-F,F.F,F -F.F./F5%)infinityG" }{TEXT -1 1 " " }}{PARA 256 "" 0 "" {TEXT -1 1 " \+ " }{XPPEDIT 18 0 "``=1/a" "6#/%!G*&\"\"\"F&%\"aG!\"\"" }{TEXT -1 1 ", " }}{PARA 0 "" 0 "" {TEXT -1 21 "so that (v) becomes: " }}{PARA 256 " " 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "Int(x^n*exp(-a*x),x = 0 .. infin ity)=n!/a^(n+1)" "6#/-%$IntG6$*&)%\"xG%\"nG\"\"\"-%$expG6#,$*&%\"aGF+F )F+!\"\"F+/F);\"\"!%)infinityG*&-%*factorialG6#F*F+)F1,&F*F+F+F+F2" } {TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 140 "The following Maple code uses a \"subsitution trick\" to avoid the occurrence of tildes \"~\" (which indicate assumed variable s) in the result. " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 " > " 0 "" {MPLTEXT 1 0 149 "assume(a_>0):\nassume(n_,posint):\nInt(x^n* exp(-a*x),x=0..infinity);\nconvert(subs(\{n_=n,a_=a\},value(subs(\{n=n _,a=a_\},%))),factorial);\na_='a_': n_='n_':" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%$IntG6$*&)%\"xG%\"nG\"\"\"-%$expG6#,$*&%\"aGF*F(F*!\" \"F*/F(;\"\"!%)infinityG" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#*()%\"aG% \"nG!\"\"F%F'-%*factorialG6#F&\"\"\"" }}}{PARA 0 "" 0 "" {TEXT -1 0 " " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{EXCHG {PARA 0 "> \+ " 0 "" {MPLTEXT 1 0 1 ";" }}}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 5 "Tasks" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{PARA 0 "" 0 "" {TEXT -1 38 "In questions 1 to 5 use the procedure " }{TEXT 0 8 "intparts" }{TEXT -1 10 " from the " } {TEXT 0 7 "student" }{TEXT -1 99 " package to demonstrate how the give n integral is evaluated using the integration by parts formula." }} {SECT 1 {PARA 4 "" 0 "" {TEXT -1 3 "Q1 " }}{PARA 0 "" 0 "" {TEXT -1 2 " " }{XPPEDIT 18 0 "Int(x*exp(-x),x);" "6#-%$IntG6$*&%\"xG\"\"\"-%$ex pG6#,$F'!\"\"F(F'" }{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 27 "____ _______________________" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" } }{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 " " {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 27 "___________________________" }}{EXCHG {PARA 0 " > " 0 "" {MPLTEXT 1 0 1 ";" }}}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 3 "Q2 " }}{PARA 0 "" 0 "" {TEXT -1 2 " " }{XPPEDIT 18 0 "Int(x^2*exp(-x),x )" "6#-%$IntG6$*&%\"xG\"\"#-%$expG6#,$F'!\"\"\"\"\"F'" }{TEXT -1 1 " \+ " }}{PARA 0 "" 0 "" {TEXT -1 27 "___________________________" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" } }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 27 "____________ _______________" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}} {SECT 1 {PARA 4 "" 0 "" {TEXT -1 3 "Q3 " }}{PARA 0 "" 0 "" {TEXT -1 2 " " }{XPPEDIT 18 0 "Int(x*ln(x),x);" "6#-%$IntG6$*&%\"xG\"\"\"-%#lnG6 #F'F(F'" }{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 27 "______________ _____________" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" } }{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 " " {TEXT -1 27 "___________________________" }}{EXCHG {PARA 0 "> " 0 " " {MPLTEXT 1 0 1 ";" }}}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 3 "Q4 " }} {PARA 0 "" 0 "" {TEXT -1 2 " " }{XPPEDIT 18 0 "Int(arctan(x),x);" "6# -%$IntG6$-%'arctanG6#%\"xGF)" }{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 27 "___________________________" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 " " {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" } }}{PARA 0 "" 0 "" {TEXT -1 27 "___________________________" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{PARA 0 "" 0 "" {TEXT -1 108 "In questions 5 to 10 determine whether the given improper integral co nverges, and if it does find its value." }}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 3 "Q5 " }}{PARA 0 "" 0 "" {TEXT -1 2 " " }{XPPEDIT 18 0 "Int (1/(x^3),x = 1 .. infinity);" "6#-%$IntG6$*&\"\"\"F'*$%\"xG\"\"$!\"\"/ F);F'%)infinityG" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 27 "_____ ______________________" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" } }{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 " " {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 27 "___________________________" }}{EXCHG {PARA 0 " > " 0 "" {MPLTEXT 1 0 1 ";" }}}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 3 "Q6 " }}{PARA 0 "" 0 "" {TEXT -1 2 " " }{XPPEDIT 18 0 "Int(x*exp(-x^2),x = 0 .. infinity);" "6#-%$IntG6$*&%\"xG\"\"\"-%$expG6#,$*$F'\"\"#!\"\" F(/F';\"\"!%)infinityG" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 27 "___________________________" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 " " {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" } }}{PARA 0 "" 0 "" {TEXT -1 27 "___________________________" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 3 "Q7 " }}{PARA 0 "" 0 "" {TEXT -1 2 " " }{XPPEDIT 18 0 "Int (x/(1+x^2),x = 0 .. infinity);" "6#-%$IntG6$*&%\"xG\"\"\",&F(F(*$F'\" \"#F(!\"\"/F';\"\"!%)infinityG" }{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 27 "___________________________" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 27 "___________________________" }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{SECT 1 {PARA 4 "" 0 " " {TEXT -1 3 "Q8 " }}{PARA 0 "" 0 "" {TEXT -1 2 " " }{XPPEDIT 18 0 "I nt(x/(1+x^4),x = 0 .. infinity);" "6#-%$IntG6$*&%\"xG\"\"\",&F(F(*$F' \"\"%F(!\"\"/F';\"\"!%)infinityG" }{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 27 "___________________________" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 27 "___________________________" }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{SECT 1 {PARA 4 "" 0 " " {TEXT -1 3 "Q9 " }}{PARA 0 "" 0 "" {TEXT -1 2 " " }{XPPEDIT 18 0 "I nt(x^3*exp(-x),x = 0 .. infinity);" "6#-%$IntG6$*&%\"xG\"\"$-%$expG6#, $F'!\"\"\"\"\"/F';\"\"!%)infinityG" }{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 27 "___________________________" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 27 "___________________________" }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{SECT 1 {PARA 4 "" 0 " " {TEXT -1 4 "Q10 " }}{PARA 0 "" 0 "" {TEXT -1 2 " " }{XPPEDIT 18 0 " Int(exp(-x^2),x = 0 .. infinity);" "6#-%$IntG6$-%$expG6#,$*$%\"xG\"\"# !\"\"/F+;\"\"!%)infinityG" }{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT 260 4 "Note" }{TEXT -1 2 ": " }{XPPEDIT 18 0 "f(x)=1/sqrt(Pi)" "6#/-% \"fG6#%\"xG*&\"\"\"F)-%%sqrtG6#%#PiG!\"\"" }{TEXT -1 1 " " }{XPPEDIT 18 0 "exp(-x^2)" "6#-%$expG6#,$*$%\"xG\"\"#!\"\"" }{TEXT -1 96 " is a \+ normal distribution curve of probability theory (corresponding to a st andard deviation of " }{XPPEDIT 18 0 "1/sqrt(2)" "6#*&\"\"\"F$-%%sqrtG 6#\"\"#!\"\"" }{TEXT -1 4 " ). " }}{PARA 0 "" 0 "" {TEXT -1 96 "It is \+ well-known that the total area under this bell-shaped curve is 1, whic h may be written as " }{XPPEDIT 18 0 "Int(exp(-x^2)/sqrt(Pi),x=-infini ty..infinity)=1" "6#/-%$IntG6$*&-%$expG6#,$*$%\"xG\"\"#!\"\"\"\"\"-%%s qrtG6#%#PiGF//F-;,$%)infinityGF/F8F0" }{TEXT -1 1 "." }}{PARA 0 "" 0 " " {TEXT -1 13 "By symmetry, " }{XPPEDIT 18 0 "Int(exp(-x^2)/sqrt(Pi),x = 0 .. infinity)=1/2" "6#/-%$IntG6$*&-%$expG6#,$*$%\"xG\"\"#!\"\"\"\" \"-%%sqrtG6#%#PiGF//F-;\"\"!%)infinityG*&F0F0F.F/" }{TEXT -1 1 "." }} {PARA 0 "" 0 "" {TEXT -1 27 "___________________________" }}{PARA 0 " " 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 27 "____________ _______________" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{SECT 1 {PARA 0 "" 0 "" {TEXT -1 18 "Code for pictures " }}{EXCHG {PARA 0 "> \+ " 0 "" {MPLTEXT 1 0 1 ";" }}}{SECT 1 {PARA 0 "" 0 "" {TEXT -1 36 "Code for improper integral pictures " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 459 "pts := [[1,0],op(op(1,op(1, plot(1/x^2,x=1..4,adaptive=false,numpoints=30)))),[4,0]]:\np1 := plots [polygonplot](pts,color=COLOR(RGB,.8,.75,1),style=PATCHNOGRID):\np2 := plot([[[1,0],[1,1]],[[4,0],[4,1/16]],[[-.03,1],[0,1]]],style=LINE,col or=black):\np3 := plot(1/x^2,x=0.8..5,color=black):\nt1 := plots[textp lot]([[1,-.07,`1`],[4,-.07,`R`],[5,-.05,`x`],[-.1,1.47,`y`],[-.1,1,1]] ):\nplots[display]([p1,p2,p3,t1],view=[-.1..5,-.07..1.5],labels=[x,y], tickmarks=[0,0]);" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 " > " 0 "" {MPLTEXT 1 0 425 "pts := [[0,0],op(op(1,op(1,plot(exp(-x),x=0 ..3,adaptive=false,numpoints=30)))),[3,0]]:\np1 := plots[polygonplot]( pts,color=COLOR(RGB,.8,.75,1),style=PATCHNOGRID):\np2 := plot(exp(-x), x=0..4,color=black):\np3 := plot([[3,0],[3,exp(-3)]],style=LINE,color= black):\nt1 := plots[textplot]([[3,-.07,`R`],[4,-.03,`x`],[-.07,1.17,` y`],[-.07,1,1]]):\nplots[display]([p1,p2,p3,t1],view=[-.07..4,-.07..1. 2],labels=[`x`,`y`],tickmarks=[0,0]);" }}}{PARA 0 "" 0 "" {TEXT -1 0 " " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 424 "pts := [[0,0],op(op(1,o p(1,plot(1/(1+x^2),x=0..3,adaptive=false,numpoints=30)))),[3,0]]:\np1 \+ := plots[polygonplot](pts,color=COLOR(RGB,.8,.75,1),style=PATCHNOGRID) :\np2 := plot(1/(1+x^2),x=0..4,color=black):\np3 := plot([[3,0],[3,.1] ],style=LINE,color=black):\nt1 := plots[textplot]([[3,-.07,`R`],[4,-.0 3,`x`],[-.07,1.17,`y`],[-.07,1,1]]):\nplots[display]([p1,p2,p3,t1],vie w=[-.07..4,-.07..1.2],labels=[`x`,`y`],tickmarks=[0,0]);" }}}{PARA 0 " " 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 454 "pts \+ := [[1,0],op(op(1,op(1,plot(1/x,x=1..4,adaptive=false,numpoints=30)))) ,[4,0]]:\np1 := plots[polygonplot](pts,color=COLOR(RGB,.8,.75,1),style =PATCHNOGRID):\np2 := plot([[[1,0],[1,1]],[[4,0],[4,1/4]],[[-.03,1],[0 ,1]]],style=LINE,color=black):\np3 := plot(1/x,x=0.6..5,color=black): \nt1 := plots[textplot]([[1,-.07,`1`],[4,-.07,`R`],[5,-.05,`x`],[-.1,1 .47,`y`],[-.1,1,1]]):\nplots[display]([p1,p2,p3,t1],view=[-.1..5,-.07. .1.5],labels=[x,y],tickmarks=[0,0]);" }}}{PARA 0 "" 0 "" {TEXT -1 0 " " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{EXCHG {PARA 0 "> \+ " 0 "" {MPLTEXT 1 0 1 ";" }}}}}{MARK "4 0 0" 0 }{VIEWOPTS 1 1 0 1 1 1803 1 1 1 1 }{PAGENUMBERS 0 1 2 33 1 1 }