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{SECT 0 {PARA 256 "" 0 "" {TEXT -1 24 "MAPLE Worksheet Number 9" }}
{PARA 257 "" 0 "" {TEXT -1 41 "Integrals in Calculus via Antiderivativ
es" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 8 "restart;" }}}{PARA 0 "" 0 "" {TEXT -1 14 "Recall that \+
  " }{XPPEDIT 18 0 " F(x) = int(f(x),x) " "6#/-%\"FG6#%\"xG-%$intG6$-%
\"fG6#F'F'" }{TEXT -1 10 "  means   " }{XPPEDIT 18 0 "diff(F(x),x)" "6
#-%%diffG6$-%\"FG6#%\"xGF)" }{TEXT -1 208 " = f(x) .  We call F(x) the
 \"indefinite integral\" of f(x) with respect to x.  Sometimes we also
 call F(x) the \"antiderivative\" of f(x) with respect to x.  Notice t
hat if C is constant with respect to x then  " }}{PARA 258 "" 0 "" 
{XPPEDIT 18 0 "diff(F(x)+C,x)" "6#-%%diffG6$,&-%\"FG6#%\"xG\"\"\"%\"CG
F+F*" }{TEXT -1 3 " = " }{XPPEDIT 18 0 "diff(F(x),x)" "6#-%%diffG6$-%
\"FG6#%\"xGF)" }{TEXT -1 11 "  =  f(x) ." }}{PARA 0 "" 0 "" {TEXT -1 
116 "hence the use of the word \"indefinite\" to describe F(x).  So th
e more common notation for the indefinite integral is" }}{PARA 259 "" 
0 "" {XPPEDIT 18 0 "int(f(x),x) " "6#-%$intG6$-%\"fG6#%\"xGF)" }{TEXT 
-1 12 "= F(x) + C ." }}{PARA 0 "" 0 "" {TEXT -1 223 "Later it will be \+
important for us to remember to include the integration constant  C . \+
 The MAPLE command for computing the indefinite integral is annologous
 to that for computting the derivative of f with respect to x.:   " }}
{PARA 260 "" 0 "" {TEXT -1 40 "int(expression,variable of integration)
;" }}{PARA 0 "" 0 "" {TEXT -1 98 "Of course the derivative and indefin
ite integral are \"inverse operations\" of each other, that is, " }}
{PARA 261 "" 0 "" {XPPEDIT 18 0 "diff(int(f(x),x),x)  =  f(x)   and in
t(diff(F(x),x),x)  =  F(x)" "6#3/-%%diffG6$-%$intG6$-%\"fG6#%\"xGF.F.-
F,6#F./-F)6$-F&6$-%\"FG6#F.F.F.-F76#F." }{TEXT -1 1 " " }}{PARA 0 "" 
0 "" {TEXT -1 145 "and MAPLE knows this.   As we did in Chapter 7 perf
orm the following commands, remember we haven't defined f or F to be a
ny particular functions." }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 40 "
Diff(int(f(x),x),x)=diff(int(f(x),x),x);" }}{PARA 11 "" 1 "" {XPPMATH 
20 "6#/-%%DiffG6$-%$intG6$-%\"fG6#%\"xGF-F-F*" }}}{EXCHG {PARA 0 "> " 
0 "" {MPLTEXT 1 0 40 "Int(diff(F(x),x),x)=int(diff(F(x),x),x);" }}
{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%$IntG6$-%%diffG6$-%\"FG6#%\"xGF-F-F
*" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 107 "Us
e MAPLE to compute each of the following indefinite integrals, ckeckin
g each result with differentiation." }}{EXCHG {PARA 0 "> " 0 "" 
{XPPEDIT 19 1 "int(x^3,x);" "6#-%$intG6$*$%\"xG\"\"$F'" }}{PARA 11 "" 
1 "" {XPPMATH 20 "6#,$*$)%\"xG\"\"%\"\"\"#F(F'" }}}{EXCHG {PARA 0 "> \+
" 0 "" {MPLTEXT 1 0 10 "diff(%,x);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#
*$)%\"xG\"\"$\"\"\"" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "
" {XPPEDIT 19 1 "int(sin(a*x),x);" "6#-%$intG6$-%$sinG6#*&%\"aG\"\"\"%
\"xGF+F," }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,$*&%\"aG!\"\"-%$cosG6#*&F
%\"\"\"%\"xGF+F+F&" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 10 "diff(
%,x);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%$sinG6#*&%\"aG\"\"\"%\"xGF(
" }}}{EXCHG {PARA 0 "> " 0 "" {XPPEDIT 19 1 "int(ln(x),x);" "6#-%$intG
6$-%#lnG6#%\"xGF)" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,&*&%\"xG\"\"\"-%
#lnG6#F%F&F&F%!\"\"" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 10 "diff
(%,x);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%#lnG6#%\"xG" }}}{EXCHG 
{PARA 0 "> " 0 "" {XPPEDIT 19 1 "int(exp(3*x),x);" "6#-%$intG6$-%$expG
6#*&\"\"$\"\"\"%\"xGF+F," }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,$-%$expG6
#,$%\"xG\"\"$#\"\"\"F)" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 10 "d
iff(%,x);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%$expG6#,$%\"xG\"\"$" }}
}{PARA 0 "" 0 "" {TEXT -1 312 "Notice that MAPLE left off the integrat
ion constant each time, or more accurately, MAPLE defaults to integrat
ion constant 0. Sometimes this causes MAPLE some greef. Consider the f
ollowing integration, first do it \"by hand\" and include the answer, \+
then use MAPLE to compute what is supposed to be the same answer." }}
{PARA 0 "" 0 "" {TEXT -1 0 "" }{TEXT 256 23 "By hand the answer is  " 
}{XPPEDIT 18 0 "(x^2+1)^21/21;" "6#*&,&*$)%\"xG\"\"#\"\"\"F)F)F)\"#@F*
!\"\"" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 
"> " 0 "" {MPLTEXT 1 0 44 "Int(2*x*(x^2+1)^20,x)=int(2*x*(x^2+1)^20,x)
;" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#/-%$IntG6$,$*&%\"xG\"\"\"),&*$)F)
\"\"#F*F*F*F*\"#?F*F/F),L*$)F)\"#UF*#F*\"#@*$)F)\"#SF*F**&\"#5F*)F)\"#
QF*F**&#\"$!>\"\"$F*)F)\"#OF*F**&\"$&GF*)F)\"#MF*F**&\"$p*F*)F)\"#KF*F
**&\"%%e#F*)F)\"#IF*F**&#\"&g(Q\"\"(F*)F)\"#GF*F**&\"%!p*F*)F)\"#EF*F*
*&#\"&!*>%FAF*)F)\"#CF*F**&\"&'z;F*)F)\"#AF*F**&FjnF*)F)F0F*F**&FenF*)
F)\"#=F*F**&FWF*)F)\"#;F*F**&FQF*)F)\"#9F*F**&FMF*)F)\"#7F*F**&FIF*)F)
F;F*F**&FEF*)F)\"\")F*F**&F?F*)F)\"\"'F*F**&F;F*)F)\"\"%F*F*F-F*" }}}
{PARA 0 "" 0 "" {TEXT -1 26 "Woah!  Try factoring this." }}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 20 "ans:=factor(rhs(%));" }}{PARA 12 "
" 1 "" {XPPMATH 20 "6#>%$ansG,$**)%\"xG\"\"#\"\"\",(*$)F(\"\"%F*F**&\"
\"$F*F'F*F*F0F*F*,0*$)F(\"#7F*F**&\"\"(F*)F(\"#5F*F**&\"#@F*)F(\"\")F*
F**&\"#NF*)F(\"\"'F*F**&F>F*F-F*F**&F:F*F'F*F*F6F*F*,<*$)F(\"#CF*F**&
\"#6F*)F(\"#AF*F**&\"#bF*)F(\"#?F*F**&\"$m\"F*)F(\"#=F*F**&\"$Q$F*)F(
\"#;F*F**&\"$!\\F*)F(\"#9F*F**&\"$>&F*F3F*F**&\"$1%F*F7F*F**&\"$N#F*F;
F*F**&\"$+\"F*F?F*F**&\"#JF*F-F*F**&F@F*F'F*F*F*F*F*#F*F:" }}}{PARA 0 
"" 0 "" {TEXT -1 117 "Well, this isn't much help.  Let's make sure it \+
is right by differentiating it to see if we get what we started with.
" }}{EXCHG {PARA 0 "> " 0 "" {XPPEDIT 19 1 "diff(ans,x);" "6#-%%diffG6
$%$ansG%\"xG" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#,***%\"xG\"\"\",(*$)F%
\"\"%F&F&*&\"\"$F&)F%\"\"#F&F&F,F&F&,0*$)F%\"#7F&F&*&\"\"(F&)F%\"#5F&F
&*&\"#@F&)F%\"\")F&F&*&\"#NF&)F%\"\"'F&F&*&F<F&F)F&F&*&F8F&F-F&F&F4F&F
&,<*$)F%\"#CF&F&*&\"#6F&)F%\"#AF&F&*&\"#bF&)F%\"#?F&F&*&\"$m\"F&)F%\"#
=F&F&*&\"$Q$F&)F%\"#;F&F&*&\"$!\\F&)F%\"#9F&F&*&\"$>&F&F1F&F&*&\"$1%F&
F5F&F&*&\"$N#F&F9F&F&*&\"$+\"F&F=F&F&*&\"#JF&F)F&F&*&F>F&F-F&F&F&F&F&#
F.F8*,#F&F8F&F-F&,&*$)F%F,F&F**&F>F&F%F&F&F&F/F&FAF&F&*,F`oF&F-F&F'F&,
.*$)F%FFF&F2*&\"#qF&)F%\"\"*F&F&*&\"$o\"F&)F%F4F&F&*&\"$5#F&)F%\"\"&F&
F&*&\"$S\"F&FcoF&F&*&\"#UF&F%F&F&F&FAF&F&*,F`oF&F-F&F'F&F/F&,:*$)F%\"#
BF&FD*&\"$U#F&)F%F8F&F&*&\"%+6F&)F%\"#>F&F&*&\"%))HF&)F%\"#<F&F&*&\"%3
aF&)F%\"#:F&F&*&\"%goF&)F%\"#8F&F&*&\"%GiF&FhoF&F&*&\"%gSF&F[pF&F&*&\"
%!)=F&F_pF&F&*&\"$+'F&FbpF&F&*&\"$C\"F&FcoF&F&*&F2F&F%F&F&F&F&" }}}
{EXCHG {PARA 0 "> " 0 "" {XPPEDIT 19 1 "factor(diff(ans,x));" "6#-%'fa
ctorG6#-%%diffG6$%$ansG%\"xG" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,$*&%
\"xG\"\"\"),&*$)F%\"\"#F&F&F&F&\"#?F&F+" }}}{PARA 0 "" 0 "" {TEXT -1 
43 "OK that seems to work, so what's going on? " }}{PARA 0 "" 0 "" 
{TEXT 257 94 "To answer this note what integration constant you got wh
en you did the calculation \"by hand\"? " }{TEXT -1 1 " " }}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 22 "expand((x^2+1)^21/21);" }}{PARA 12 
"" 1 "" {XPPMATH 20 "6#,N*$)%\"xG\"#U\"\"\"#F(\"#@*$)F&\"#SF(F(*&\"#5F
()F&\"#QF(F(*&#\"$!>\"\"$F()F&\"#OF(F(*&\"$&GF()F&\"#MF(F(*&\"$p*F()F&
\"#KF(F(*&\"%%e#F()F&\"#IF(F(*&#\"&g(Q\"\"(F()F&\"#GF(F(*&\"%!p*F()F&
\"#EF(F(*&#\"&!*>%F5F()F&\"#CF(F(*&\"&'z;F()F&\"#AF(F(*&FTF()F&\"#?F(F
(*&FOF()F&\"#=F(F(*&FKF()F&\"#;F(F(*&FEF()F&\"#9F(F(*&FAF()F&\"#7F(F(*
&F=F()F&F/F(F(*&F9F()F&\"\")F(F(*&F3F()F&\"\"'F(F(*&F/F()F&\"\"%F(F(*$
)F&\"\"#F(F(F)F(" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 25 "The constant \+
term is 1/21" }}}{PARA 0 "" 0 "" {TEXT -1 105 "Notice the constant ter
m in the above integral is 0.  Try adding your constant to it and fact
oring again." }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 17 "factor(ans+1
/21);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,$*$),&*$)%\"xG\"\"#\"\"\"F+F
+F+\"#@F+#F+F," }}}{PARA 0 "" 0 "" {TEXT -1 114 "Both answers are corr
ect, they represent the same indefinite integral, I personally prefer \+
the one we get by hand." }}{PARA 0 "" 0 "" {TEXT -1 95 "Now we'll do s
ome more complicated integration problems, you know, like those \"neat
\" ones from " }}{PARA 0 "" 0 "" {TEXT -1 20 "Cal II.   Try these." }}
{EXCHG {PARA 0 "> " 0 "" {XPPEDIT 19 1 "int(sin(x)^2*cos(x),x);" "6#-%
$intG6$*&-%$sinG6#%\"xG\"\"#-%$cosG6#F*\"\"\"F*" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#,$*$)-%$sinG6#%\"xG\"\"$\"\"\"#F+F*" }}}{PARA 0 "" 0 "
" {TEXT 258 61 "What integration method was used here? (Above.)  Subst
itution" }}{EXCHG {PARA 0 "> " 0 "" {XPPEDIT 19 1 "int(sin(x)^2*cos(x)
^2,x);" "6#-%$intG6$*&-%$sinG6#%\"xG\"\"#-%$cosG6#F*F+F*" }}{PARA 11 "
" 1 "" {XPPMATH 20 "6#,(*&-%$sinG6#%\"xG\"\"\")-%$cosGF'\"\"$F)#!\"\"
\"\"%*(#F)\"\")F)F+F)F%F)F)*&F2F)F(F)F)" }}}{EXCHG {PARA 0 "> " 0 "" 
{XPPEDIT 19 1 "int(x*exp(x),x);" "6#-%$intG6$*&%\"xG\"\"\"-%$expG6#F'F
(F'" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,&*&%\"xG\"\"\"-%$expG6#F%F&F&F
'!\"\"" }}}{EXCHG {PARA 0 "> " 0 "" {XPPEDIT 19 1 "int(x^3*exp(x),x);
" "6#-%$intG6$*&%\"xG\"\"$-%$expG6#F'\"\"\"F'" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#,**&)%\"xG\"\"$\"\"\"-%$expG6#F&F(F(*(F'F()F&\"\"#F(F)F
(!\"\"*(\"\"'F(F&F(F)F(F(*&F1F(F)F(F/" }}}{PARA 0 "" 0 "" {TEXT 259 
92 "What integration method was used in these last two examples? (Abov
e.)  integration by parts." }}{EXCHG {PARA 0 "> " 0 "" {XPPEDIT 19 1 "
int(1/sqrt(1-x^2),x);" "6#-%$intG6$*&\"\"\"F'-%%sqrtG6#,&F'F'*$%\"xG\"
\"#!\"\"F/F-" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%'arcsinG6#%\"xG" }}}
{EXCHG {PARA 0 "> " 0 "" {XPPEDIT 19 1 "int(x/sqrt(1-x^2),x);" "6#-%$i
ntG6$*&%\"xG\"\"\"-%%sqrtG6#,&F(F(*$F'\"\"#!\"\"F/F'" }}{PARA 11 "" 1 
"" {XPPMATH 20 "6#,$*$-%%sqrtG6#,&\"\"\"F)*$)%\"xG\"\"#F)!\"\"F)F." }}
}{EXCHG {PARA 0 "> " 0 "" {XPPEDIT 19 1 "int(x^3/sqrt(1-x^2),x);" "6#-
%$intG6$*&%\"xG\"\"$-%%sqrtG6#,&\"\"\"F-*$F'\"\"#!\"\"F0F'" }}{PARA 
11 "" 1 "" {XPPMATH 20 "6#,&*&)%\"xG\"\"#\"\"\"-%%sqrtG6#,&F(F(*$F%F(!
\"\"F(#F.\"\"$*&#F'F0F(*$F)F(F(F." }}}{PARA 0 "" 0 "" {TEXT 260 175 "W
hat integration method(s) was(were) used in these last two examples? (
Above.)  first one was substitution x=sqrt(1-x^2).  It seems likely th
e second used the same substituion" }}{PARA 0 "" 0 "" {TEXT -1 126 "We
 can compute double integrals just as easily as we computed mixed deri
vatives.  Compute the following multiple integration: " }}{EXCHG 
{PARA 0 "> " 0 "" {XPPEDIT 19 1 "int(int(x*y+x^3-y^2+2,x),y);" "6#-%$i
ntG6$-F$6$,**&%\"xG\"\"\"%\"yGF+F+*$F*\"\"$F+*$F,\"\"#!\"\"F0F+F*F," }
}{PARA 11 "" 1 "" {XPPMATH 20 "6#,**&)%\"xG\"\"#\"\"\")%\"yGF'F(#F(\"
\"%*(F+F()F&F,F(F*F(F(*&#F(\"\"$F(*&)F*F1F(F&F(F(!\"\"*(F'F(F&F(F*F(F(
" }}}{PARA 0 "" 0 "" {TEXT -1 67 "Compute the same integral with the o
rder of integration reversed.  " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 
1 0 28 "int(int(x*y+x^3-y^2+2,y),x);" }}{PARA 11 "" 1 "" {XPPMATH 20 "
6#,**&)%\"xG\"\"#\"\"\")%\"yGF'F(#F(\"\"%*(F+F()F&F,F(F*F(F(*&#F(\"\"$
F(*&)F*F1F(F&F(F(!\"\"*(F'F(F&F(F*F(F(" }}}{PARA 0 "" 0 "" {TEXT 261 
37 "What do you observe?  They are equal." }}{PARA 0 "" 0 "" {TEXT -1 
232 "Does the order of integration ever make any difference?  Actually
 in definite integration sometimes it make a lot of difference when tr
ying to compute \"by hand.\".  Try to compute  the following double in
tegration problem by hand:   " }{XPPEDIT 18 0 "int(int(exp(x^2),x=y..1
),y=0..1)" "6#-%$intG6$-F$6$-%$expG6#*$%\"xG\"\"#/F,;%\"yG\"\"\"/F0;\"
\"!F1" }{TEXT -1 4 "  . " }{TEXT 262 93 "What is the problem you incou
nter immediately?   I don't know an antiderivative for exp(x^2) " }}
{PARA 265 "" 0 "" {TEXT -1 214 "Now reverse the order of integration a
nd compute the same integral.  (Remember you must draw the region of i
ntegration and choose the appropriate limits of integration in the new
 order.)    The correct integral is " }{XPPEDIT 18 0 "Int(Int(e^(x^2),
y = 0 .. x),x = 0 .. 1);" "6#-%$IntG6$-F$6$*$)%\"eG*$)%\"xG\"\"#\"\"\"
F//%\"yG;\"\"!%\"xG/F4;F3F/" }}{PARA 0 "" 0 "" {TEXT -1 74 "Now comput
e the integral in both orders with MAPLE. Include both results. " }}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 34 "\nint(int(exp(x^2),x=y..1),y
=0..1);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,&-%$expG6#\"\"\"#F'\"\"##F
'F)!\"\"" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 42 "\nint(int(exp(x
^2),y = 0 .. x),x = 0 .. 1);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,&-%$e
xpG6#\"\"\"#F'\"\"##F'F)!\"\"" }}}{PARA 0 "" 0 "" {TEXT 263 20 "What d
o you observe?" }{TEXT -1 15 "   Equal again." }}{PARA 0 "" 0 "" 
{TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 202 "There are lots of contin
uous functions which simply don't have a \"closed form antiderivative.
\"  By this we mean there are functions f(x) for which there is no for
mula expression in x,  F(x), for which  " }}{PARA 0 "" 0 "" {TEXT -1 
1 " " }{XPPEDIT 18 0 "diff(F(x),x)=f(x)" "6#/-%%diffG6$-%\"FG6#%\"xGF*
-%\"fG6#F*" }{TEXT -1 52 "  . We have already seen one such function, \+
namely, " }{XPPEDIT 18 0 "exp(x^2) " "6#-%$expG6#*$%\"xG\"\"#" }{TEXT 
-1 14 " ,another is  " }{XPPEDIT 18 0 "sin(x^2)" "6#-%$sinG6#*$%\"xG\"
\"#" }{TEXT -1 81 " . Ask MAPLE to compute the indefinite integral of \+
both and display the result.  " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 
0 16 "int(exp(x^2),x);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#*(^##!\"\"\"
\"#\"\"\"-%%sqrtG6#%#PiGF(-%$erfG6#*&^#F(F(%\"xGF(F(" }}}{EXCHG {PARA 
0 "> " 0 "" {MPLTEXT 1 0 16 "int(sin(x^2),x);" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#,$*(-%%sqrtG6#\"\"#\"\"\"-F&6#%#PiGF)-%)FresnelSG6#*(F(
#F)F(F,#!\"\"F(%\"xGF)F)F1" }}}{PARA 0 "" 0 "" {TEXT -1 85 "Of course \+
the Fundamental Theorem tells us there is a function whose derivative \+
is   " }{XPPEDIT 18 0 "sin(x^2)" "6#-%$sinG6#*$%\"xG\"\"#" }{TEXT -1 
13 "  ,namely    " }{XPPEDIT 18 0 "int(sin(t^2),t=0..x)" "6#-%$intG6$-
%$sinG6#*$%\"tG\"\"#/F*;\"\"!%\"xG" }{TEXT -1 201 "  is such a functio
n, but we would have to use some numerical integration technique to co
mpute its value for particular values of x.  For example use the trape
zoidal rule to compute the value for x=5. " }}{EXCHG {PARA 0 "> " 0 "
" {MPLTEXT 1 0 14 "with(student):" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 38 "trapezoid(sin(t^2), t=0..5); evalf(%);" }}{PARA 11 "
" 1 "" {XPPMATH 20 "6#,&-%$SumG6$-%$sinG6#,$*$)%\"iG\"\"#\"\"\"#\"#D\"
#;/F-;F/\"\"$#\"\"&\"\"%*&#F7\"\")F/-F(6#F1F/F/" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#$\"+m-GsB!\"*" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}
{PARA 0 "" 0 "" {TEXT -1 88 "There are other integrals that MAPLE need
s some help in solving.  For example ask MAPLE " }}{PARA 0 "" 0 "" 
{TEXT -1 12 "to compute  " }{XPPEDIT 18 0 "int(h,x)" "6#-%$intG6$%\"hG
%\"xG" }{TEXT -1 31 "  for the function h defined as" }}{EXCHG {PARA 
0 "> " 0 "" {XPPEDIT 19 1 "h:=1/sqrt(1+sqrt(x));" "6#>%\"hG*&\"\"\"F&-
%%sqrtG6#,&F&F&-F(6#%\"xGF&!\"\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%
\"hG*&\"\"\"F&*$-%%sqrtG6#,&F&F&*$-F)6#%\"xGF&F&F&!\"\"" }}}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 262 "" 1 "" {TEXT -1 65 "
Load the student package and try the following command sequence: " }}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "
" {MPLTEXT 1 0 36 "changevar(1+sqrt(x)=u^2,Int(h,x),u);" }}{PARA 11 "
" 1 "" {XPPMATH 20 "6#-%$IntG6$,$*$-%%sqrtG6#,(\"\"\"F,*&\"\"#F,)%\"uG
F.F,!\"\"*$)F0\"\"%F,F,F,F4F0" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 
0 9 "value(%);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,$***$),&!\"\"\"\"\"
*$)%\"uG\"\"#F)F)F-F)#F)F-F,F),&F*F)\"\"$F(F)F'F(#\"\"%F0" }}}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 31 "ans:=subs(u=sqrt(1+sqrt(x)),%);" }}
{PARA 11 "" 1 "" {XPPMATH 20 "6#>%$ansG,$*&-%%sqrtG6#,&\"\"\"F+*$-F(6#
%\"xGF+F+F+,&!\"#F+F,F+F+#\"\"%\"\"$" }}}{PARA 0 "" 0 "" {TEXT -1 43 "
We can check the answer by differentiating." }}{EXCHG {PARA 0 "> " 0 "
" {MPLTEXT 1 0 12 "diff(ans,x);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,&*
(,&\"\"\"F&*$-%%sqrtG6#%\"xGF&F&#!\"\"\"\"#,&!\"#F&F'F&F&F+F,#F&\"\"$*
(#F.F2F&F%#F&F.F+F,F&" }}}{PARA 0 "" 0 "" {TEXT -1 44 "This doesn't lo
ok right, try simplifying it." }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 
0 12 "simplify(%);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#*&\"\"\"F$*$-%%s
qrtG6#,&F$F$*$-F'6#%\"xGF$F$F$!\"\"" }}}{PARA 0 "" 0 "" {TEXT -1 25 "N
ow ask MAPLE to compute " }{XPPEDIT 18 0 "int(g,x)" "6#-%$intG6$%\"gG%
\"xG" }{TEXT -1 31 "  for the function g defined by" }}{EXCHG {PARA 0 
"> " 0 "" {XPPEDIT 19 1 "g:=ln(x+sqrt(1+x^2));" "6#>%\"gG-%#lnG6#,&%\"
xG\"\"\"-%%sqrtG6#,&F*F**$F)\"\"#F*F*" }}{PARA 11 "" 1 "" {XPPMATH 20 
"6#>%\"gG-%#lnG6#,&%\"xG\"\"\"*$-%%sqrtG6#,&F*F**$)F)\"\"#F*F*F*F*" }}
}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 9 "int(g,x);" }}{PARA 11 "" 1 "
" {XPPMATH 20 "6#,&*&-%#lnG6#,&%\"xG\"\"\"*$-%%sqrtG6#,&F*F**$)F)\"\"#
F*F*F*F*F*F)F*F*F+!\"\"" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 45 "Well, \+
not really, see if the following works:" }}}{PARA 0 "" 0 "" {TEXT -1 
87 "Again, MAPLE needs help.  This time we'll try integration by parts
.  Recall the formula" }}{PARA 263 "" 0 "" {XPPEDIT 18 0 "int(u,v)" "6
#-%$intG6$%\"uG%\"vG" }{TEXT -1 3 " = " }{XPPEDIT 18 0 "uv-int(v,u)" "
6#,&%#uvG\"\"\"-%$intG6$%\"vG%\"uG!\"\"" }{TEXT -1 2 " ." }}{PARA 0 "
" 0 "" {TEXT -1 36 "Our only option seems to be to let  " }{XPPEDIT 
18 0 "u = ln(x+sqrt(1+x^2))" "6#/%\"uG-%#lnG6#,&%\"xG\"\"\"-%%sqrtG6#,
&F*F**$F)\"\"#F*F*" }{TEXT -1 12 "    and     " }{XPPEDIT 18 0 "dv=dx
" "6#/%#dvG%#dxG" }{TEXT -1 21 "  . Try the following" }}{PARA 0 "" 0 
"" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 186 "Try simplifying this \+
and computing the value of the result to see if MAPLE can handle it no
w. Of course you are to check your answer by differentiation (and simp
lification if necessary)." }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 37 
"intparts(Int(g,x),ln(x+sqrt(1+x^2)));" }}{PARA 11 "" 1 "" {XPPMATH 
20 "6#,&*&-%#lnG6#,&%\"xG\"\"\"*$-%%sqrtG6#,&F*F**$)F)\"\"#F*F*F*F*F*F
)F*F*-%$IntG6$*(,&F*F**&F/#!\"\"F2F)F*F*F*F(F:F)F*F)F:" }}}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 12 "simplify(%);" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#,&*&-%#lnG6#,&%\"xG\"\"\"*$-%%sqrtG6#,&F*F**$)F)\"\"#F*
F*F*F*F*F)F*F*-%$IntG6$*&F/#!\"\"F2F)F*F)F8" }}}{EXCHG {PARA 0 "> " 0 
"" {MPLTEXT 1 0 12 "G:=value(%);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%
\"GG,&*&-%#lnG6#,&%\"xG\"\"\"*$-%%sqrtG6#,&F,F,*$)F+\"\"#F,F,F,F,F,F+F
,F,F-!\"\"" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 10 "diff(G,x);" }
}{PARA 11 "" 1 "" {XPPMATH 20 "6#,(*(,&\"\"\"F&*&,&F&F&*$)%\"xG\"\"#F&
F&#!\"\"F,F+F&F&F&,&F+F&*$-%%sqrtG6#F(F&F&F.F+F&F&-%#lnG6#F/F&F'F." }}
}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 12 "simplify(%);" }}{PARA 11 "
" 1 "" {XPPMATH 20 "6#-%#lnG6#,&%\"xG\"\"\"*$-%%sqrtG6#,&F(F(*$)F'\"\"
#F(F(F(F(" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 9 "It works." }}}{PARA 
264 "" 1 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 33 "Use the same \+
method to compute   " }{XPPEDIT 18 0 "int((arcsin(x))^2,x)" "6#-%$intG
6$*$-%'arcsinG6#%\"xG\"\"#F*" }{TEXT -1 49 "  .  Again  check you answ
er via differentiation." }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 30 "o
urint := Int(arcsin(x)^2, x);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%'ou
rintG-%$IntG6$*$)-%'arcsinG6#%\"xG\"\"#\"\"\"F-" }}}{EXCHG {PARA 0 "> \+
" 0 "" {MPLTEXT 1 0 9 "value(%);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%
$intG6$*$)-%'arcsinG6#%\"xG\"\"#\"\"\"F+" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 42 "secondint :=intparts(ourint, arcsin(x)^2);" }}{PARA 
11 "" 1 "" {XPPMATH 20 "6#>%*secondintG,&*&)-%'arcsinG6#%\"xG\"\"#\"\"
\"F+F-F--%$IntG6$,$*(F(F-,&F-F-*$)F+F,F-!\"\"#F6F,F+F-F,F+F6" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 17 "value(secondint);" }}{PARA 
11 "" 1 "" {XPPMATH 20 "6#,&*&)-%'arcsinG6#%\"xG\"\"#\"\"\"F)F+F+-%$in
tG6$,$*(F&F+,&F+F+*$)F)F*F+!\"\"#F4F*F)F+F*F)F4" }}}{EXCHG {PARA 0 "" 
0 "" {TEXT -1 31 "Try integration by parts again." }}}{EXCHG {PARA 0 "
> " 0 "" {MPLTEXT 1 0 31 "intparts(secondint, arcsin(x));" }}{PARA 11 
"" 1 "" {XPPMATH 20 "6#,(*&)-%'arcsinG6#%\"xG\"\"#\"\"\"F)F+F+*(F*F+F&
F+-%%sqrtG6#,&F+F+*$)F)F*F+!\"\"F+F+-%$IntG6$!\"#F)F+" }}}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 12 "H:=value(%);" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#>%\"HG,(*&)-%'arcsinG6#%\"xG\"\"#\"\"\"F+F-F-*(F,F-F(F-
-%%sqrtG6#,&F-F-*$)F+F,F-!\"\"F-F-*&F,F-F+F-F5" }}}{EXCHG {PARA 0 "> \+
" 0 "" {MPLTEXT 1 0 10 "diff(H,x);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#
*$)-%'arcsinG6#%\"xG\"\"#\"\"\"" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 9 
"it works." }}}{PARA 11 "" 1 "" {TEXT -1 0 "" }}}{MARK "109 0 0" 31 }
{VIEWOPTS 1 1 0 1 1 1803 1 1 1 1 }{PAGENUMBERS 0 1 2 33 1 1 }