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{SECT 0 {EXCHG {PARA 18 "" 0 "" {TEXT -1 11 "Calculus II" }}{PARA 257 
"" 0 "" {TEXT -1 32 "Lesson 10:  Integration by Parts" }}}{SECT 0 
{PARA 3 "" 0 "" {TEXT -1 29 "Anti-differentiation by Parts" }}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 8 "restart;" }}}{EXCHG {PARA 0 "" 0 "" 
{TEXT -1 64 "The second main method of anti-differentiation we will st
udy is " }{TEXT 256 29 "anti-differentiation by parts" }{TEXT -1 59 ".
  As discussed in class, this is summarised by the formula" }}{PARA 
256 "" 0 "" {XPPEDIT 18 0 "Int(u*D(v),x) = u*v-Int(v*D(u),x);" "6#/-%$
IntG6$*&%\"uG\"\"\"-%\"DG6#%\"vGF)%\"xG,&*&F(F)F-F)F)-F%6$*&F-F)-F+6#F
(F)F.!\"\"" }{TEXT -1 2 " ." }}{PARA 0 "" 0 "" {TEXT 257 5 "Maple" }
{TEXT -1 60 " has a command which will integrate by parts.  It is call
ed " }{TEXT 258 8 "intparts" }{TEXT -1 24 ", and it resides in the " }
{TEXT 259 7 "student" }{TEXT -1 51 " package, which we will have to re
load because the " }{TEXT 260 7 "restart" }{TEXT -1 30 " above cleared
 it from memory." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 14 "with(st
udent);" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#7@%\"DG%%DiffG%*DoubleintG%
$IntG%&LimitG%(LineintG%(ProductG%$SumG%*TripleintG%*changevarG%/compl
etesquareG%)distanceG%'equateG%*integrandG%*interceptG%)intpartsG%(lef
tboxG%(leftsumG%)makeprocG%*middleboxG%*middlesumG%)midpointG%(powsubs
G%)rightboxG%)rightsumG%,showtangentG%(simpsonG%&slopeG%(summandG%*tra
pezoidG" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 63 "(Note that we suppress
ed the list of commands this time.)  The " }{TEXT 261 8 "intparts" }
{TEXT -1 116 " command has two arguments: the first is the expression \+
to be anti-differentiated, and the second is the choice for " }
{XPPEDIT 18 0 "u;" "6#%\"uG" }{TEXT -1 69 ", the piece which is to be \+
differentiated.  We don't need to specify " }{XPPEDIT 18 0 "D(v);" "6#
-%\"DG6#%\"vG" }{TEXT -1 54 ", because it must be everything else in t
he integrand." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" 
{TEXT -1 26 "As an example, let's find " }{XPPEDIT 18 0 "Int(x*exp(-2*
x),x);" "6#-%$IntG6$*&%\"xG\"\"\"-%$expG6#,$*&\"\"#F(F'F(!\"\"F(F'" }
{TEXT -1 73 ".  Hopefully, you can look at this example and see that w
e should choose " }{XPPEDIT 18 0 "u = x;" "6#/%\"uG%\"xG" }{TEXT -1 1 
"." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 25 "p1 := Int(x*exp(-2*x)
,x);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#p1G-%$IntG6$*&%\"xG\"\"\"-%
$expG6#,$F)!\"#F*F)" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 21 "p2 :
= intparts(p1,x);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#p2G,&*&%\"xG\"
\"\"-%$expG6#,$F'!\"#F(#!\"\"\"\"#-%$IntG6$,$F)F.F'F/" }}}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 19 "p3 := simplify(p2);" }}{PARA 11 "" 
1 "" {XPPMATH 20 "6#>%#p3G,&*&%\"xG\"\"\"-%$expG6#,$F'!\"#F(#!\"\"\"\"
#*&#F(F0F(-%$IntG6$F)F'F(F(" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 120 "W
e still have an integral to do, but it is sufficiently simple that we \+
can see how to evaluate it, and so we will allow " }{TEXT 262 5 "Maple
" }{TEXT -1 10 " to do so:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 
16 "p4 := value(p3);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#p4G,&*&%\"x
G\"\"\"-%$expG6#,$F'!\"#F(#!\"\"\"\"#*&#F(\"\"%F(F)F(F/" }}}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 12 "p1 = p4 + C;" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#/-%$IntG6$*&%\"xG\"\"\"-%$expG6#,$F(!\"#F)F(,(F'#!\"\"
\"\"#*&#F)\"\"%F)F*F)F1%\"CGF)" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 34 
"Finally, check by differentiation:" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 11 "diff(p4,x);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#*&%\"x
G\"\"\"-%$expG6#,$F$!\"#F%" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 16 "Dif
ferentiating " }{TEXT 263 2 "p4" }{TEXT -1 40 " gives the function we \+
started with, so " }{TEXT 264 2 "p4" }{TEXT -1 41 " is an anti-derivat
ive for this function." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 
0 "" {TEXT -1 47 "There are some nice tricks you can do with the " }
{TEXT 265 8 "intparts" }{TEXT -1 76 " command.  For example, a standar
d use of the method of parts is to compute " }{XPPEDIT 18 0 "Int(ln(x)
,x);" "6#-%$IntG6$-%#lnG6#%\"xGF)" }{TEXT -1 18 " by writing it as " }
{XPPEDIT 18 0 "Int(1*ln(x),x);" "6#-%$IntG6$*&\"\"\"F'-%#lnG6#%\"xGF'F
+" }{TEXT -1 14 " and choosing " }{XPPEDIT 18 0 "u = ln(x);" "6#/%\"uG
-%#lnG6#%\"xG" }{TEXT -1 1 "." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 
0 19 "p1 := Int(ln(x),x);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#p1G-%$
IntG6$-%#lnG6#%\"xGF+" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 25 "p2
 := intparts(p1,ln(x));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#p2G,&*&-
%#lnG6#%\"xG\"\"\"F*F+F+-%$IntG6$F+F*!\"\"" }}}{EXCHG {PARA 0 "> " 0 "
" {MPLTEXT 1 0 16 "p3 := value(p2);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6
#>%#p3G,&*&-%#lnG6#%\"xG\"\"\"F*F+F+F*!\"\"" }}}{EXCHG {PARA 0 "> " 0 
"" {MPLTEXT 1 0 12 "p1 = p3 + C;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-
%$IntG6$-%#lnG6#%\"xGF*,(*&F'\"\"\"F*F-F-F*!\"\"%\"CGF-" }}}{EXCHG 
{PARA 0 "" 0 "" {TEXT -1 46 "Of course, we should check by differentia
tion:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 11 "diff(p3,x);" }}
{PARA 11 "" 1 "" {XPPMATH 20 "6#-%#lnG6#%\"xG" }}}{EXCHG {PARA 0 "> " 
0 "" {MPLTEXT 1 0 0 "" }}}{SECT 0 {PARA 4 "" 0 "" {TEXT -1 10 "Questio
n 2" }}{EXCHG {PARA 0 "" 0 "" {TEXT -1 204 "Use the method described a
bove to find the following anti-derivatives.  (All of them are taken f
rom Exercises 7.1 of Stewart, and you should do them all---and others-
--with pencil and paper for homework.)" }}}{EXCHG {PARA 0 "" 0 "" 
{TEXT -1 4 "(a) " }{XPPEDIT 18 0 "Int(x*cos(x),x);" "6#-%$IntG6$*&%\"x
G\"\"\"-%$cosG6#F'F(F'" }{TEXT -1 2 "\n\n" }{TEXT 266 9 "Solution." }
{TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 22 "p1 := Int(x
*cos(x),x);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#p1G-%$IntG6$*&%\"xG
\"\"\"-%$cosG6#F)F*F)" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 21 "p2
 := intparts(p1,x);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#p2G,&*&%\"xG
\"\"\"-%$sinG6#F'F(F(-%$IntG6$F)F'!\"\"" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 16 "p3 := value(p2);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>
%#p3G,&*&%\"xG\"\"\"-%$sinG6#F'F(F(-%$cosGF+F(" }}}{EXCHG {PARA 0 "> \+
" 0 "" {MPLTEXT 1 0 12 "p1 = p3 + C;" }}{PARA 11 "" 1 "" {XPPMATH 20 "
6#/-%$IntG6$*&%\"xG\"\"\"-%$cosG6#F(F)F(,(*&F(F)-%$sinGF,F)F)F*F)%\"CG
F)" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 12 "Check: (You " }{TEXT 267 3 
"did" }{TEXT -1 41 " check each of your answers, didn't you?)" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 11 "diff(p3,x);" }}{PARA 11 "" 
1 "" {XPPMATH 20 "6#*&%\"xG\"\"\"-%$cosG6#F$F%" }}}{EXCHG {PARA 0 "> \+
" 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 4 "(b) " }
{XPPEDIT 18 0 "Int(x*ln(x),x);" "6#-%$IntG6$*&%\"xG\"\"\"-%#lnG6#F'F(F
'" }{TEXT -1 2 "\n\n" }{TEXT 268 9 "Solution." }{TEXT -1 0 "" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 22 "p1 := Int(x*ln(x), x);" }}
{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#p1G-%$IntG6$*&-%#lnG6#%\"xG\"\"\"F,
F-F," }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 26 "p2 := intparts(p1, \+
ln(x));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#p2G,&*&-%#lnG6#%\"xG\"\"
\")F*\"\"#F+#F+F--%$IntG6$,$F*F.F*!\"\"" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 19 "p3 := simplify(p2);" }}{PARA 11 "" 1 "" {XPPMATH 20 "
6#>%#p3G,&*&-%#lnG6#%\"xG\"\"\")F*\"\"#F+#F+F-*&#F+F-F+-%$IntG6$F*F*F+
!\"\"" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 16 "p4 := value(p3);" 
}}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#p4G,&*&-%#lnG6#%\"xG\"\"\")F*\"\"
#F+#F+F-*&#F+\"\"%F+*$F,F+F+!\"\"" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 12 "p1 = p4 + C;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%$I
ntG6$*&-%#lnG6#%\"xG\"\"\"F+F,F+,(*&F(F,)F+\"\"#F,#F,F0*&#F,\"\"%F,*$F
/F,F,!\"\"%\"CGF," }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 6 "Check:" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 11 "diff(p4,x);" }}{PARA 11 "" 
1 "" {XPPMATH 20 "6#*&-%#lnG6#%\"xG\"\"\"F'F(" }}}{EXCHG {PARA 0 "> " 
0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 4 "(c) " }
{XPPEDIT 18 0 "Int(x^2*sin(2*x),x);" "6#-%$IntG6$*&%\"xG\"\"#-%$sinG6#
*&F(\"\"\"F'F-F-F'" }{TEXT -1 2 "\n\n" }{TEXT 269 9 "Solution." }
{TEXT -1 59 "  It seems clear that we should differentiate the power o
f " }{XPPEDIT 18 0 "x;" "6#%\"xG" }{TEXT -1 173 ".  This will lower th
e power by 1, which will give us a simpler integral, but not yet one t
hat we are allowed to evaluate.  We will have to integrate by parts a \+
second time." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 29 "p1 := Int(x
^2 * sin(2*x), x);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#p1G-%$IntG6$*
&)%\"xG\"\"#\"\"\"-%$sinG6#,$F*F+F,F*" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 24 "p2 := intparts(p1, x^2);" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#>%#p2G,&*&)%\"xG\"\"#\"\"\"-%$cosG6#,$F(F)F*#!\"\"F)-%$
IntG6$,$*&F(F*F+F*F0F(F0" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 19 
"p3 := simplify(p2);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#p3G,&*&)%\"
xG\"\"#\"\"\"-%$cosG6#,$F(F)F*#!\"\"F)-%$IntG6$*&F(F*F+F*F(F*" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 22 "p4 := intparts(p3, x);" }}
{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#p4G,(*&)%\"xG\"\"#\"\"\"-%$cosG6#,$
F(F)F*#!\"\"F)*(#F*F)F*F(F*-%$sinGF-F*F*-%$IntG6$,$F3F2F(F0" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 19 "p5 := simplify(p4);" }}
{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#p5G,(*&)%\"xG\"\"#\"\"\"-%$cosG6#,$
F(F)F*#!\"\"F)*(#F*F)F*F(F*-%$sinGF-F*F**&#F*F)F*-%$IntG6$F3F(F*F0" }}
}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 16 "p6 := value(p5);" }}{PARA 
11 "" 1 "" {XPPMATH 20 "6#>%#p6G,(*&)%\"xG\"\"#\"\"\"-%$cosG6#,$F(F)F*
#!\"\"F)*(#F*F)F*F(F*-%$sinGF-F*F**&#F*\"\"%F*F+F*F*" }}}{EXCHG {PARA 
0 "> " 0 "" {MPLTEXT 1 0 12 "p1 = p6 + C;" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#/-%$IntG6$*&)%\"xG\"\"#\"\"\"-%$sinG6#,$F)F*F+F),**&F(F
+-%$cosGF.F+#!\"\"F**(#F+F*F+F)F+F,F+F+*&#F+\"\"%F+F2F+F+%\"CGF+" }}}
{EXCHG {PARA 0 "" 0 "" {TEXT -1 6 "Check:" }}}{EXCHG {PARA 0 "> " 0 "
" {MPLTEXT 1 0 11 "diff(p6,x);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#*&)%
\"xG\"\"#\"\"\"-%$sinG6#,$F%F&F'" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 4 "(d) " }
{XPPEDIT 18 0 "Int(x^2*exp(-x),x);" "6#-%$IntG6$*&%\"xG\"\"#-%$expG6#,
$F'!\"\"\"\"\"F'" }{TEXT -1 2 "\n\n" }{TEXT 270 9 "Solution." }{TEXT 
-1 68 "  This is the same as part (c): we have to integrate by parts t
wice." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 28 "p1 := Int(x^2 * ex
p(-x), x);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#p1G-%$IntG6$*&)%\"xG
\"\"#\"\"\"-%$expG6#,$F*!\"\"F,F*" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 24 "p2 := intparts(p1, x^2);" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#>%#p2G,&*&)%\"xG\"\"#\"\"\"-%$expG6#,$F(!\"\"F*F/-%$Int
G6$,$*&F(F*F+F*!\"#F(F/" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 19 "
p3 := simplify(p2);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#p3G,&*&)%\"x
G\"\"#\"\"\"-%$expG6#,$F(!\"\"F*F/*&F)F*-%$IntG6$*&F(F*F+F*F(F*F*" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 22 "p4 := intparts(p3, x);" }}
{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#p4G,(*&)%\"xG\"\"#\"\"\"-%$expG6#,$
F(!\"\"F*F/*(F)F*F(F*F+F*F/*&F)F*-%$IntG6$,$F+F/F(F*F/" }}}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 19 "p5 := simplify(p4);" }}{PARA 11 "" 
1 "" {XPPMATH 20 "6#>%#p5G,(*&)%\"xG\"\"#\"\"\"-%$expG6#,$F(!\"\"F*F/*
(F)F*F(F*F+F*F/*&F)F*-%$IntG6$F+F(F*F*" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 16 "p6 := value(p5);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>
%#p6G,(*&)%\"xG\"\"#\"\"\"-%$expG6#,$F(!\"\"F*F/*(F)F*F(F*F+F*F/*&F)F*
F+F*F/" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 12 "p1 = p6 + C;" }}
{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%$IntG6$*&)%\"xG\"\"#\"\"\"-%$expG6#
,$F)!\"\"F+F),*F'F0*(F*F+F)F+F,F+F0*&F*F+F,F+F0%\"CGF+" }}}{EXCHG 
{PARA 0 "" 0 "" {TEXT -1 6 "Check:" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 12 "diff(p6, x);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#*&)%
\"xG\"\"#\"\"\"-%$expG6#,$F%!\"\"F'" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 0 "" }}}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}
{SECT 0 {PARA 4 "" 0 "" {TEXT -1 10 "Question 3" }}{EXCHG {PARA 0 "" 
0 "" {TEXT -1 122 "Find the following anti-derivatives.  You will prob
ably have to use a combination of both methods: substitution and parts
." }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 4 "(a) " }{XPPEDIT 18 0 "Int(x^3
*exp(x^2),x);" "6#-%$IntG6$*&%\"xG\"\"$-%$expG6#*$F'\"\"#\"\"\"F'" }
{TEXT -1 2 "\n\n" }{TEXT 271 9 "Solution." }{TEXT -1 98 "  Whenever yo
u have an exponential function in an integral, it is a good idea to tr
y substituting " }{XPPEDIT 18 0 "u;" "6#%\"uG" }{TEXT -1 42 " for what
ever function is in the exponent." }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 29 "p1 := Int(x^3 * exp(x^2), x);" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#>%#p1G-%$IntG6$*&)%\"xG\"\"$\"\"\"-%$expG6#*$)F*\"\"#F,
F,F*" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 30 "p2 := changevar(u=x
^2, p1, u);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#p2G-%$IntG6$,$*&%\"u
G\"\"\"-%$expG6#F*F+#F+\"\"#F*" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 
1 0 19 "p3 := simplify(p2);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#p3G,
$-%$IntG6$*&%\"uG\"\"\"-%$expG6#F*F+F*#F+\"\"#" }}}{EXCHG {PARA 0 "> \+
" 0 "" {MPLTEXT 1 0 22 "p4 := intparts(p3, u);" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#>%#p4G,&*&%\"uG\"\"\"-%$expG6#F'F(#F(\"\"#*&#F(F-F(-%$I
ntG6$F)F'F(!\"\"" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 16 "p5 := v
alue(p4);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#p5G,&*&%\"uG\"\"\"-%$e
xpG6#F'F(#F(\"\"#*&#F(F-F(F)F(!\"\"" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 22 "p6 := subs(u=x^2, p5);" }}{PARA 11 "" 1 "" {XPPMATH 
20 "6#>%#p6G,&*&)%\"xG\"\"#\"\"\"-%$expG6#*$F'F*F*#F*F)*&#F*F)F*F+F*!
\"\"" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 12 "p1 = p6 + C;" }}
{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%$IntG6$*&)%\"xG\"\"$\"\"\"-%$expG6#
*$)F)\"\"#F+F+F),(*&F0F+F,F+#F+F1*&#F+F1F+F,F+!\"\"%\"CGF+" }}}{EXCHG 
{PARA 0 "" 0 "" {TEXT -1 6 "Check:" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 12 "diff(p6, x);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#*&)%
\"xG\"\"$\"\"\"-%$expG6#*$)F%\"\"#F'F'" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 4 "(b) " }
{XPPEDIT 18 0 "Int(cos(x)*ln(sin(x)),x);" "6#-%$IntG6$*&-%$cosG6#%\"xG
\"\"\"-%#lnG6#-%$sinG6#F*F+F*" }{TEXT -1 2 "\n\n" }{TEXT 272 9 "Soluti
on." }{TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 32 "p1 :=
 Int(cos(x)*ln(sin(x)), x);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#p1G-
%$IntG6$*&-%$cosG6#%\"xG\"\"\"-%#lnG6#-%$sinGF+F-F," }}}{EXCHG {PARA 
0 "> " 0 "" {MPLTEXT 1 0 33 "p2 := changevar(u=sin(x), p1, u);" }}
{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#p2G-%$IntG6$-%#lnG6#%\"uGF+" }}}
{EXCHG {PARA 0 "" 0 "" {TEXT -1 198 "We found this anti-derivative ear
lier on in the worksheet, so we might allow ourselves to evaluate it a
s it stands, but for practice (and because it is not in the list on p.
 416), let's do it again." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 
26 "p3 := intparts(p2, ln(u));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#p
3G,&*&-%#lnG6#%\"uG\"\"\"F*F+F+-%$IntG6$F+F*!\"\"" }}}{EXCHG {PARA 0 "
> " 0 "" {MPLTEXT 1 0 16 "p4 := value(p3);" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#>%#p4G,&*&-%#lnG6#%\"uG\"\"\"F*F+F+F*!\"\"" }}}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 25 "p5 := subs(u=sin(x), p4);" }}{PARA 
11 "" 1 "" {XPPMATH 20 "6#>%#p5G,&*&-%#lnG6#-%$sinG6#%\"xG\"\"\"F*F.F.
F*!\"\"" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 12 "p1 = p5 + C;" }}
{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%$IntG6$*&-%$cosG6#%\"xG\"\"\"-%#lnG
6#-%$sinGF*F,F+,(*&F-F,F0F,F,F0!\"\"%\"CGF," }}}{EXCHG {PARA 0 "" 0 "
" {TEXT -1 6 "Check:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 12 "dif
f(p5, x);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#*&-%$cosG6#%\"xG\"\"\"-%#
lnG6#-%$sinGF&F(" }}}}}}{MARK "0 1 0" 9 }{VIEWOPTS 1 1 0 1 1 1803 1 1 
1 1 }{PAGENUMBERS 0 1 2 33 1 1 }