{VERSION 6 0 "IBM INTEL NT" "6.0" } {USTYLETAB {CSTYLE "Maple Input" -1 0 "Courier" 0 1 255 0 0 1 0 1 0 0 1 0 0 0 0 1 }{CSTYLE "2D Math" -1 2 "Times" 0 1 0 0 0 0 0 0 2 0 0 0 0 0 0 1 }{CSTYLE "2D Comment" 2 18 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 1 } {CSTYLE "" -1 256 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 257 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 258 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 259 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 260 "" 1 12 0 0 0 0 0 0 0 0 0 0 0 0 0 1 } {CSTYLE "" -1 261 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 262 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 263 "" 1 12 0 0 0 0 0 0 0 0 0 0 0 0 0 0 }{PSTYLE "Normal" -1 0 1 {CSTYLE "" -1 -1 "Tim es" 1 12 0 0 0 1 2 2 2 2 2 2 1 1 1 1 }1 1 0 0 0 0 1 0 1 0 2 2 0 1 } {PSTYLE "Heading 1" -1 3 1 {CSTYLE "" -1 -1 "Times" 1 18 0 0 0 1 2 1 2 2 2 2 1 1 1 1 }1 1 0 0 6 6 1 0 1 0 2 2 0 1 }{PSTYLE "Bullet Item" -1 15 1 {CSTYLE "" -1 -1 "Times" 1 12 0 0 0 1 2 2 2 2 2 2 1 1 1 1 }1 1 0 0 3 3 1 0 1 0 2 2 15 2 }{PSTYLE "Heading 1" -1 256 1 {CSTYLE "" -1 -1 "Times" 1 18 0 0 0 1 2 1 2 2 2 2 1 1 1 1 }3 1 0 0 6 6 1 0 1 0 2 2 0 1 }{PSTYLE "Heading 2" -1 257 1 {CSTYLE "" -1 -1 "Times" 1 12 0 0 0 1 2 1 2 2 2 2 1 1 1 1 }3 1 0 0 4 4 1 0 1 0 2 2 0 1 }{PSTYLE "Heading 2" -1 258 1 {CSTYLE "" -1 -1 "Times" 1 14 0 0 0 1 2 1 2 2 2 2 1 1 1 1 }3 1 0 0 4 4 1 0 1 0 2 2 0 1 }} {SECT 0 {PARA 256 "" 0 "" {TEXT -1 25 "Techniques of Integration" }} {PARA 257 "" 0 "" {TEXT -1 9 "Jim Herod" }}{PARA 257 "" 0 "" {TEXT -1 12 "P O Box 1038" }}{PARA 257 "" 0 "" {TEXT -1 25 "Grove Hill, Alabama 36451" }}{PARA 258 "" 0 "" {TEXT 260 19 "herod@math.gatech.e" }{TEXT 263 2 "du" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 136 "It used to be that calculus texts asked students to study techniq ues of integration so well that they could evaluate an integral such a s" }}{PARA 0 "" 0 "" {TEXT -1 32 " " } {XPPEDIT 18 0 "int(arcsin(t),t=0..x)" "6#-%$intG6$-%'arcsinG6#%\"tG/F) ;\"\"!%\"xG" }{TEXT -1 22 ", for 0 < x < 1." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 138 "Such integrals might be \+ relegated to a computer, now. One might guess that Maple will do this \+ handily. It's true. Consider the following: " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 22 "int(arcsin(t),t=0. .x);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 " " {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 36 " This raises severa l questions: " }}{PARA 15 "" 0 "" {TEXT -1 116 "How can a human unders tand the techniques for computing the value of this integral? What are appropriate techniques?" }}{PARA 15 "" 0 "" {TEXT -1 66 "Can Maple al so integrate powers of arcsin(x)? And, if not, can we " }{TEXT 256 5 " teach" }{TEXT -1 18 " Maple to do this?" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{SECT 1 {PARA 3 "" 0 "" {TEXT -1 49 "Integration Technique: The M ethod of Substitution" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 139 " Because of the success in having Maple to integr ate arcsin(t), it is irresistible to try having Maple integrate the sq uare of arcsine." }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 24 "int(arcs in(t)^2,t=0..x);" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 160 " The success of Maple in integrating this latter mak es the challenge greater. We begin by reading in the package \"student \" which contains many useful ideas." }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 14 "with(student):" }}}{PARA 0 "" 0 "" {TEXT -1 44 "The i ntegral we wish to learn to evaluate is" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 17 "Int(arcsin(x),x);" }}}{PARA 0 "" 0 "" {TEXT -1 42 "Th e idea will be to make the substitution " }{TEXT 258 13 "u = arcsin(x) " }{TEXT -1 16 ". In that case, " }{TEXT 257 10 "x = sin(u)" }{TEXT -1 5 " and " }{TEXT 259 14 "dx = cos(u) du" }{TEXT -1 137 ". It is imp ortant to recognize that for u's in the range of arcsin(x), the functi on cos(u) is positive. To see this, think of the graphs." }}{PARA 0 " " 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 36 "plot( \{sin(x),cos(x)\},x=-Pi/2..Pi/2);" }}}{PARA 0 "" 0 "" {TEXT -1 38 "Thu s, we make the important assumption" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 17 "assume(cos(u)>0);" }}}{PARA 0 "" 0 "" {TEXT -1 36 "No w, we make the change of variables" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 42 "changevar(arcsin(x)=u,Int(arcsin(x),x),u);" }}}{PARA 0 "" 0 "" {TEXT -1 57 "This integral is not hard to evaluate; essentia lly it is " }}{PARA 0 "" 0 "" {TEXT -1 40 " \+ " }{XPPEDIT 18 0 "int(u*cos(u),u)" "6#-%$intG6$*&%\"uG\" \"\"-%$cosG6#F'F(F'" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 34 "Th is could be integrated by parts." }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 19 "value(simplify(%));" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 20 "subs(u=arcsin(x),%);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 12 "simplify(%);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 85 " Thus, we h ave a procedure for integrating arcsin(x) that humans can understand. \+ " }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 261 9 "Prob lem1:" }{TEXT -1 41 " Revise the above procedure to integrate" }} {PARA 0 "" 0 "" {TEXT -1 38 " " } {XPPEDIT 18 0 "int(arcsin(x)^2,x)" "6#-%$intG6$*$-%'arcsinG6#%\"xG\"\" #F*" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{SECT 1 {PARA 3 "" 0 "" {TEXT -1 44 "Integration Technique: Integration-by-parts " } }{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 250 "An alte rnate method of integration is the technique of integration-by-parts. \+ A proof of this technique can be made by recalling the formula for the derivative of two functions multiplied together. With this recollecti on, it is not so hard to see that" }}{PARA 0 "" 0 "" {TEXT -1 23 " \+ " }{XPPEDIT 18 0 "f(b)*g(b)-f(a)*g(a) = Int(f(x)*di ff(g(x),x),x=a..b)+Int(diff(f(x),x)*g(x),x=a..b)" "6#/,&*&-%\"fG6#%\"b G\"\"\"-%\"gG6#F)F*F**&-F'6#%\"aGF*-F,6#F1F*!\"\",&-%$IntG6$*&-F'6#%\" xGF*-%%diffG6$-F,6#F6$-F'6#F " 0 "" {MPLTEXT 1 0 14 "with(student):" }}}{EXCHG {PARA 0 " > " 0 "" {MPLTEXT 1 0 69 "Int(f(x)*diff(g(x),x), x)=\nintparts(Int(f(x )*diff(g(x),x), x), f(x));" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 41 "We use this idea to evaluate the integral" }} {PARA 0 "" 0 "" {TEXT -1 38 " " } {XPPEDIT 18 0 "int(x*arcsin(x),x)" "6#-%$intG6$*&%\"xG\"\"\"-%'arcsinG 6#F'F(F'" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 39 "intparts(Int(x*arcsin(x),x),arcsin( x));" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 19 "value(simplify(%)); " }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 162 "Thi s technique of integration-by-parts is often used multiple times in th e evaluation of one integral. The example that all students remember i s the evaluation of" }}{PARA 0 "" 0 "" {TEXT -1 26 " \+ " }{XPPEDIT 18 0 "int(exp(x)*sin(x),x)" "6#-%$intG6$*&-%$expG6 #%\"xG\"\"\"-%$sinG6#F*F+F*" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 40 "intparts(Int(exp(x )*sin(x), x), exp(x));" }}}{PARA 0 "" 0 "" {TEXT -1 64 "We evaluate th is last integral using integration-by-parts again." }}{EXCHG {PARA 0 " > " 0 "" {MPLTEXT 1 0 55 "-exp(x)*cos(x)+intparts(Int(exp(x)*cos(x), x ), exp(x));" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 262 9 "Problem2:" }{TEXT -1 41 " Revise the above procedur e to integrate" }}{PARA 0 "" 0 "" {TEXT -1 38 " \+ " }{XPPEDIT 18 0 "int(x^2*arcsin(x),x)" "6#-%$intG6$*&% \"xG\"\"#-%'arcsinG6#F'\"\"\"F'" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{MARK "0 0" 0 }{VIEWOPTS 1 1 0 1 1 1803 1 1 1 1 } {PAGENUMBERS 0 1 2 33 1 1 }