{VERSION 4 0 "IBM INTEL NT" "4.0" }
{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 "2D Output" 2 20 "" 0 1 0 0 255 1 0 0 0 0 0 0 0 0 0 1 }
{CSTYLE "" -1 256 "" 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 1 }{CSTYLE "" -1 
257 "" 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 1 }{CSTYLE "" -1 258 "" 0 1 0 0 
0 0 0 0 1 0 0 0 0 0 0 1 }{CSTYLE "" -1 259 "" 0 1 0 0 0 0 0 0 1 0 0 0 
0 0 0 1 }{CSTYLE "" -1 260 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 1 }
{CSTYLE "" -1 261 "" 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 1 }{CSTYLE "" 18 
262 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 0 }{CSTYLE "" 18 263 "" 0 1 0 0 
0 0 0 1 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 264 "" 0 1 0 0 0 0 0 1 0 0 0 0 
0 0 0 0 }{PSTYLE "Normal" -1 0 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 0 0 1 0 1 0 2 2 0 1 }{PSTYLE "Maple Out
put" -1 11 1 {CSTYLE "" -1 -1 "Times" 1 12 0 0 0 1 2 2 2 2 2 2 1 1 1 
1 }3 3 0 0 0 0 1 0 1 0 2 2 0 1 }{PSTYLE "Maple Output" -1 12 1 
{CSTYLE "" -1 -1 "Times" 1 12 0 0 0 1 2 2 2 2 2 2 1 1 1 1 }1 3 0 0 0 
0 1 0 1 0 2 2 0 1 }{PSTYLE "List Item" -1 14 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 14 5 }
{PSTYLE "Title" -1 18 1 {CSTYLE "" -1 -1 "Times" 1 18 0 0 0 1 2 1 1 2 
2 2 1 1 1 1 }3 1 0 0 12 12 1 0 1 0 2 2 19 1 }{PSTYLE "Normal" -1 256 
1 {CSTYLE "" -1 -1 "Times" 1 12 0 0 0 1 2 1 2 2 2 2 1 1 1 1 }1 1 0 0 
0 0 1 0 1 0 2 2 0 1 }{PSTYLE "Heading 2" -1 257 1 {CSTYLE "" -1 -1 "Ti
mes" 1 14 0 0 0 1 2 1 2 2 2 2 1 1 1 1 }3 1 0 0 8 2 1 0 1 0 2 2 0 1 }}
{SECT 0 {EXCHG {PARA 18 "" 0 "" {TEXT -1 11 "Calculus II" }}{PARA 257 
"" 0 "" {TEXT -1 55 "Lesson 9:  Integration by Substitution: Worked Ex
amples" }}}{EXCHG {PARA 14 "" 0 "" {TEXT -1 160 "Here are 20 anti-deri
vatives.  We reduce each of them to a simpler form by means of a subst
itution.  Here are the rules for today's game:\n\n Use the unevaluated
 " }{TEXT 256 3 "Int" }{TEXT -1 318 " command throughout the worksheet
.\n\nYou may consider that you have \"solved\" the problem when you ha
ve reduced it to one of the standard forms or to a problem which can b
e done with a known reduction formula, or to a sum of problems of thes
e types.\n\nTry to find the simplest substitution that will work in ea
ch case.  \n" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 8 "restart;" }}
}{EXCHG {PARA 14 "" 0 "" {TEXT -1 21 "Remember to load the " }{TEXT 
257 7 "student" }{TEXT -1 30 " package, so that you can use " }{TEXT 
258 9 "changevar" }{TEXT -1 2 " ." }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 
14 "with(student);" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#7@%\"DG%%DiffG%*
DoubleintG%$IntG%&LimitG%(LineintG%(ProductG%$SumG%*TripleintG%*change
varG%/completesquareG%)distanceG%'equateG%*integrandG%*interceptG%)int
partsG%(leftboxG%(leftsumG%)makeprocG%*middleboxG%*middlesumG%)midpoin
tG%(powsubsG%)rightboxG%)rightsumG%,showtangentG%(simpsonG%&slopeG%(su
mmandG%*trapezoidG" }}}{EXCHG {PARA 256 "" 0 "" {TEXT -1 4 "1.  " }
{XPPEDIT 18 0 "Int(sqrt(9-x^2)/(x^2),x);" "6#-%$IntG6$*&-%%sqrtG6#,&\"
\"*\"\"\"*$%\"xG\"\"#!\"\"F,*$F.F/F0F." }{TEXT -1 0 "" }}}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 32 "p1 := Int(sqrt(9 - x^2)/x^2, x);" }
}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#p1G-%$IntG6$*&,&\"\"*\"\"\"*$)%\"x
G\"\"#F+!\"\"#F+F/F.!\"#F." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 
35 "p2 := changevar(x=3*sin(u), p1, u);" }}{PARA 11 "" 1 "" {XPPMATH 
20 "6#>%#p2G-%$IntG6$,$*(,&\"\"*\"\"\"*&F+F,)-%$sinG6#%\"uG\"\"#F,!\"
\"#F,F3F/!\"#-%$cosGF1F,#F,\"\"$F2" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 
-1 15 "Unfortunately, " }{TEXT 259 8 "simplify" }{TEXT -1 30 " doesn't
 do what we want here:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 13 "s
implify(p2);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,$-%$IntG6$*(-%%csgnG6
#-%$cosG6#%\"uG\"\"\"F+\"\"#,&!\"\"F/*$)F+F0F/F/F2F.F2" }}}{EXCHG 
{PARA 0 "" 0 "" {TEXT -1 19 "The reason is that " }{TEXT 260 5 "Maple
" }{TEXT -1 19 " will not simplify " }{XPPEDIT 18 0 "sqrt(a^2);" "6#-%
%sqrtG6#*$%\"aG\"\"#" }{TEXT -1 4 " as " }{XPPEDIT 18 0 "a;" "6#%\"aG
" }{TEXT -1 22 " unless it knows that " }{XPPEDIT 18 0 "a;" "6#%\"aG" 
}{TEXT -1 46 " is positive.  We must tell it to assume that " }
{XPPEDIT 18 0 "cos(u);" "6#-%$cosG6#%\"uG" }{TEXT -1 19 " is positive;
 then " }{TEXT 261 8 "simplify" }{TEXT -1 54 " works.  (Why are we all
owed to make this assumption?)" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 
1 0 20 "assume(cos(u) >= 0);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 
0 19 "p3 := simplify(p2);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#p3G,$-
%$IntG6$*&-%$cosG6#%#u|irG\"\"#,&!\"\"\"\"\"*$)F*F.F1F1F0F-F0" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 42 "p4 := subs(-1 + cos(u)^2 = -
sin(u)^2, p3);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#p4G,$-%$IntG6$,$*
&-%$cosG6#%#u|irG\"\"#-%$sinGF-!\"#!\"\"F.F3" }}}{EXCHG {PARA 0 "> " 
0 "" {MPLTEXT 1 0 47 "p5 := subs(cos(u)^2 / sin(u)^2 = cot(u)^2, p4);
" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#p5G,$-%$IntG6$,$*$)-%$cotG6#%#u
|irG\"\"#\"\"\"!\"\"F/F2" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 40 
"p6 := subs(cot(u)^2 = csc(u)^2 - 1, p5);" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#>%#p6G,$-%$IntG6$,&*$)-%$cscG6#%#u|irG\"\"#\"\"\"!\"\"F
1F1F/F2" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 26 "We are down to evaluat
ing " }{XPPEDIT 18 0 "Int(csc(u)^2,u);" "6#-%$IntG6$*$-%$cscG6#%\"uG\"
\"#F*" }{TEXT -1 5 " and " }{XPPEDIT 18 0 "Int(1,u);" "6#-%$IntG6$\"\"
\"%\"uG" }{TEXT -1 1 " " }}}{EXCHG {PARA 256 "" 0 "" {TEXT -1 4 "2.  \+
" }{XPPEDIT 18 0 "Int(1/(x^2*sqrt(x^2+4)),x);" "6#-%$IntG6$*&\"\"\"F'*
&%\"xG\"\"#-%%sqrtG6#,&*$F)F*F'\"\"%F'F'!\"\"F)" }{TEXT -1 0 "" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 38 "p1 := Int(1/(x^2 * sqrt(x^2 \+
+ 4)), x);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#p1G-%$IntG6$*&\"\"\"F
)*&)%\"xG\"\"#F)-%%sqrtG6#,&*$F+F)F)\"\"%F)F)!\"\"F," }}}{EXCHG {PARA 
0 "> " 0 "" {MPLTEXT 1 0 37 "p2 := changevar(x = 2*tan(u), p1, u);" }}
{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#p2G-%$IntG6$,$*(,&\"\"\"F+*$)-%$tan
G6#%#u|irG\"\"#F+F+F+F.!\"#,&F,\"\"%F5F+#!\"\"F2#F+F2F1" }}}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 19 "p3 := simplify(p2);" }}{PARA 11 "" 
1 "" {XPPMATH 20 "6#>%#p3G,$-%$IntG6$*&,&\"\"\"F+*$)-%$tanG6#%#u|irG\"
\"#F+F+#F+F2F.!\"#F1#F+\"\"%" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 264 3 "3
. " }{XPPEDIT 263 0 "Int(x^2/sqrt(x^2+4),x);" "6#-%$IntG6$*&%\"xG\"\"#
-%%sqrtG6#,&*$F'F(\"\"\"\"\"%F.!\"\"F'" }{TEXT -1 0 "" }}}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 34 "p1 := Int(x^2 / sqrt(x^2 + 4), x);
" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#p1G-%$IntG6$*&%\"xG\"\"#,&*$)F)
F*\"\"\"F.\"\"%F.#!\"\"F*F)" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 
37 "p2 := changevar(x = 2*tan(u), p1, u);" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#>%#p2G-%$IntG6$,$*(-%$tanG6#%#u|irG\"\"#,&\"\"\"F0*$)F*
F.F0F0F0,&F1\"\"%F4F0#!\"\"F.\"\")F-" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 19 "p3 := simplify(p2);" }}{PARA 11 "" 1 "" {XPPMATH 20 "
6#>%#p3G,$-%$IntG6$*&)-%$tanG6#%#u|irG\"\"#\"\"\"-%%sqrtG6#,&F0F0*$F*F
0F0F0F.\"\"%" }}}{EXCHG {PARA 256 "" 0 "" {TEXT -1 4 "4.  " }{XPPEDIT 
18 0 "Int(x/sqrt(x^2+4),x);" "6#-%$IntG6$*&%\"xG\"\"\"-%%sqrtG6#,&*$F'
\"\"#F(\"\"%F(!\"\"F'" }{TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 32 "p1 := Int(x / sqrt(x^2 + 4), x);" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#>%#p1G-%$IntG6$*&%\"xG\"\"\",&*$)F)\"\"#F*F*\"\"%F*#!\"
\"F.F)" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 34 "p2 := changevar(u
=x^2 + 4, p1, u);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#p2G-%$IntG6$,$
*&\"\"\"F**$-%%sqrtG6#%#u|irGF*!\"\"#F*\"\"#F/" }}}{EXCHG {PARA 0 "> \+
" 0 "" {MPLTEXT 1 0 19 "p3 := simplify(p2);" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#>%#p3G,$-%$IntG6$*&\"\"\"F**$-%%sqrtG6#%#u|irGF*!\"\"F/
#F*\"\"#" }}}{EXCHG {PARA 256 "" 0 "" {TEXT -1 4 "5.  " }{XPPEDIT 18 
0 "Int(1/sqrt(x^2-4),x);" "6#-%$IntG6$*&\"\"\"F'-%%sqrtG6#,&*$%\"xG\"
\"#F'\"\"%!\"\"F0F-" }{TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 30 "p1 := Int(1/sqrt(x^2 - 4), x);" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#>%#p1G-%$IntG6$*&\"\"\"F)*$-%%sqrtG6#,&*$)%\"xG\"\"#F)F
)\"\"%!\"\"F)F4F1" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 37 "p2 := \+
changevar(x = 2*sec(u), p1, u);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#
p2G-%$IntG6$,$*(,&*$)-%$secG6#%#u|irG\"\"#\"\"\"\"\"%F3!\"\"#F4F1F-F2-
%$tanGF/F2F1F0" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 19 "p3 := sim
plify(p2);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#p3G-%$IntG6$*(-%$cosG
6#%#u|irG!\"\",&\"\"\"F/*$)F)\"\"#F/F-#F-F2-%$sinGF+F/F," }}}{EXCHG 
{PARA 256 "" 0 "" {TEXT -1 4 "6.  " }{XPPEDIT 262 0 "Int(sec(x)*tan(x)
*sqrt(1+sec(x)),x);" "6#-%$IntG6$*(-%$secG6#%\"xG\"\"\"-%$tanG6#F*F+-%
%sqrtG6#,&F+F+-F(6#F*F+F+F*" }{TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "
" {MPLTEXT 1 0 46 "p1 := Int( sec(x)*tan(x)*sqrt(1 + sec(x)), x);" }}
{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#p1G-%$IntG6$*(-%$secG6#%\"xG\"\"\"-
%$tanGF+F--%%sqrtG6#,&F-F-F)F-F-F," }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 39 "p2 := changevar(u = 1 + sec(x), p1, u);" }}{PARA 11 "
" 1 "" {XPPMATH 20 "6#>%#p2G-%$IntG6$*$-%%sqrtG6#%#u|irG\"\"\"F," }}}
{EXCHG {PARA 256 "" 0 "" {TEXT -1 4 "7.  " }{XPPEDIT 18 0 "Int(x*sqrt(
3*x+1),x);" "6#-%$IntG6$*&%\"xG\"\"\"-%%sqrtG6#,&*&\"\"$F(F'F(F(F(F(F(
F'" }{TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 30 "p1 := \+
Int(x*sqrt(3*x + 1), x);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#p1G-%$I
ntG6$*&%\"xG\"\"\"-%%sqrtG6#,&F)\"\"$F*F*F*F)" }}}{EXCHG {PARA 0 "> " 
0 "" {MPLTEXT 1 0 35 "p2 := changevar(u = 3*x+ 1, p1, u);" }}{PARA 11 
"" 1 "" {XPPMATH 20 "6#>%#p2G-%$IntG6$,$*&,&#!\"\"\"\"$\"\"\"*&#F.F-F.
%#u|irGF.F.F.-%%sqrtG6#F1F.F0F1" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 
1 0 19 "p3 := simplify(p2);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#p3G,
$-%$IntG6$*&,&!\"\"\"\"\"%#u|irGF,F,-%%sqrtG6#F-F,F-#F,\"\"*" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 256 "" 0 "
" {TEXT -1 4 "8.  " }{XPPEDIT 18 0 "Int(x^3*sqrt(9-x^2),x);" "6#-%$Int
G6$*&%\"xG\"\"$-%%sqrtG6#,&\"\"*\"\"\"*$F'\"\"#!\"\"F.F'" }{TEXT -1 0 
"" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 34 "p1 := Int(x^3 * sqrt(9
 - x^2), x);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#p1G-%$IntG6$*&)%\"x
G\"\"$\"\"\"-%%sqrtG6#,&\"\"*F,*$)F*\"\"#F,!\"\"F,F*" }}}{EXCHG {PARA 
0 "> " 0 "" {MPLTEXT 1 0 36 "p2 := changevar(u = 9 - x^2, p1, u);" }}
{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#p2G-%$IntG6$,$*&,&\"\"*\"\"\"%#u|ir
G!\"\"F,-%%sqrtG6#F-F,#F.\"\"#F-" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 19 "p3 := simplify(p2);" }}{PARA 11 "" 1 "" {XPPMATH 20 "
6#>%#p3G,$-%$IntG6$*&,&!\"*\"\"\"%#u|irGF,F,-%%sqrtG6#F-F,F-#F,\"\"#" 
}}}{EXCHG {PARA 256 "" 0 "" {TEXT -1 4 "9.  " }{XPPEDIT 18 0 "Int(cos(
x)^4*sin(x),x);" "6#-%$IntG6$*&-%$cosG6#%\"xG\"\"%-%$sinG6#F*\"\"\"F*
" }{TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 32 "p1 := In
t(cos(x)^4 * sin(x), x);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#p1G-%$I
ntG6$*&)-%$cosG6#%\"xG\"\"%\"\"\"-%$sinGF,F/F-" }}}{EXCHG {PARA 0 "> \+
" 0 "" {MPLTEXT 1 0 35 "p2 := changevar(u = cos(x), p1, u);" }}{PARA 
11 "" 1 "" {XPPMATH 20 "6#>%#p2G-%$IntG6$,$*$)%#u|irG\"\"%\"\"\"!\"\"F
+" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 19 "p3 := simplify(p2);" }
}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#p3G,$-%$IntG6$*$)%#u|irG\"\"%\"\"
\"F+!\"\"" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG 
{PARA 256 "" 0 "" {TEXT -1 4 "10. " }{XPPEDIT 18 0 "Int(1/(((x+1)^2+1)
^2),x);" "6#-%$IntG6$*&\"\"\"F'*$,&*$,&%\"xGF'F'F'\"\"#F'F'F'F-!\"\"F,
" }{TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 33 "p1 := In
t(1/( (x+1)^2 + 1)^2, x);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#p1G-%$
IntG6$*&\"\"\"F)*$),&*$),&%\"xGF)F)F)\"\"#F)F)F)F)F1F)!\"\"F0" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 37 "p2 := changevar(x+1 = tan(u)
, p1, u);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#p2G-%$IntG6$*&\"\"\"F)
,&F)F)*$)-%$tanG6#%#u|irG\"\"#F)F)!\"\"F0" }}}{EXCHG {PARA 256 "" 0 "
" {TEXT -1 4 "11. " }{XPPEDIT 18 0 "Int(x^3*sqrt(4-9*x^2),x);" "6#-%$I
ntG6$*&%\"xG\"\"$-%%sqrtG6#,&\"\"%\"\"\"*&\"\"*F.*$F'\"\"#F.!\"\"F.F'
" }{TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 36 "p1 := In
t(x^3 * sqrt(4 - 9*x^2), x);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#p1G
-%$IntG6$*&)%\"xG\"\"$\"\"\"-%%sqrtG6#,&\"\"%F,*&\"\"*F,)F*\"\"#F,!\"
\"F,F*" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 38 "p2 := changevar(u
 = 4 - 9*x^2, p1, u);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#p2G-%$IntG
6$,$*&,&#\"\"%\"\"*\"\"\"*&#F.F-F.%#u|irGF.!\"\"F.-%%sqrtG6#F1F.#F2\"#
=F1" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 19 "p3 := simplify(p2);
" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#p3G,$-%$IntG6$*&,&!\"%\"\"\"%#u
|irGF,F,-%%sqrtG6#F-F,F-#F,\"$i\"" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 256 "" 0 "" {TEXT -1 4 "12. " }
{XPPEDIT 18 0 "Int(x*sqrt(1+x^4),x);" "6#-%$IntG6$*&%\"xG\"\"\"-%%sqrt
G6#,&F(F(*$F'\"\"%F(F(F'" }{TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 30 "p1 := Int(x*sqrt(1 + x^4), x);" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#>%#p1G-%$IntG6$*&%\"xG\"\"\"-%%sqrtG6#,&F*F**$)F)\"\"%F
*F*F*F)" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 32 "p2 := changevar(
u = x^2, p1, u);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#p2G-%$IntG6$,$*
$-%%sqrtG6#,&\"\"\"F.*$)%#u|irG\"\"#F.F.F.#F.F2F1" }}}{EXCHG {PARA 0 "
> " 0 "" {MPLTEXT 1 0 19 "p3 := simplify(p2);" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#>%#p3G,$-%$IntG6$*$-%%sqrtG6#,&\"\"\"F.*$)%#u|irG\"\"#F
.F.F.F1#F.F2" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 35 "p4 := chang
evar(u = tan(t), p3, t);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#p4G,$-%
$IntG6$*$),&\"\"\"F,*$)-%$tanG6#%\"tG\"\"#F,F,#\"\"$F3F,F2#F,F3" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 40 "p5 := subs(1 + tan(t)^2 = se
c(t)^2, p4);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#p5G,$-%$IntG6$*$)*$
)-%$secG6#%\"tG\"\"#\"\"\"#\"\"$F1F2F0#F2F1" }}}{EXCHG {PARA 0 "> " 0 
"" {MPLTEXT 1 0 19 "assume(sec(t) > 0);" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 26 "p6 := simplify(p5, power);" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#>%#p6G,$-%$IntG6$*$)-%$secG6#%#t|irG\"\"$\"\"\"F.#F0\"
\"#" }}}{EXCHG {PARA 256 "" 0 "" {TEXT -1 4 "13. " }{XPPEDIT 18 0 "Int
(x^5*(x^3+1)^(1/3),x);" "6#-%$IntG6$*&%\"xG\"\"&),&*$F'\"\"$\"\"\"F-F-
*&F-F-F,!\"\"F-F'" }{TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 36 "p1 := Int(x^5 * (x^3 + 1)^(1/3), x);" }}{PARA 11 "" 
1 "" {XPPMATH 20 "6#>%#p1G-%$IntG6$*&)%\"xG\"\"&\"\"\"),&*$)F*\"\"$F,F
,F,F,#F,F1F,F*" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 36 "p2 := cha
ngevar(u = x^3 + 1, p1, u);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#p2G-
%$IntG6$,$*&,&!\"\"\"\"\"%#u|irGF,F,)F-#F,\"\"$F,F/F-" }}}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 19 "p3 := simplify(p2);" }}{PARA 11 "" 
1 "" {XPPMATH 20 "6#>%#p3G,$-%$IntG6$*&,&!\"\"\"\"\"%#u|irGF,F,)F-#F,
\"\"$F,F-F/" }}}{EXCHG {PARA 256 "" 0 "" {TEXT -1 4 "14. " }{XPPEDIT 
18 0 "Int(x/sqrt(1+2*x),x);" "6#-%$IntG6$*&%\"xG\"\"\"-%%sqrtG6#,&F(F(
*&\"\"#F(F'F(F(!\"\"F'" }{TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 30 "p1 := Int(x/sqrt(1 + 2*x), x);" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#>%#p1G-%$IntG6$*&%\"xG\"\"\",&F*F**&\"\"#F*F)F*F*#!\"\"
F-F)" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 36 "p2 := changevar(u =
 1 + 2*x, p1, u);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#p2G-%$IntG6$,$
*&,&#!\"\"\"\"#\"\"\"*&#F.F-F.%#u|irGF.F.F.F1F+F0F1" }}}{EXCHG {PARA 
0 "> " 0 "" {MPLTEXT 1 0 19 "p3 := simplify(p2);" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#>%#p3G,$-%$IntG6$*&,&!\"\"\"\"\"%#u|irGF,F,F-#F+\"\"#F-
#F,\"\"%" }}}{EXCHG {PARA 256 "" 0 "" {TEXT -1 4 "15. " }{XPPEDIT 18 
0 "Int(exp(x)*sqrt(9-exp(2*x)),x);" "6#-%$IntG6$*&-%$expG6#%\"xG\"\"\"
-%%sqrtG6#,&\"\"*F+-F(6#*&\"\"#F+F*F+!\"\"F+F*" }{TEXT -1 0 "" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 41 "p1 := Int( exp(x)*sqrt(9 - e
xp(2*x)), x);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#p1G-%$IntG6$*&-%$e
xpG6#%\"xG\"\"\"-%%sqrtG6#,&\"\"*F--F*6#,$F,\"\"#!\"\"F-F," }}}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 35 "p2 := changevar(u = exp(x), p1, u);
" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#p2G-%$IntG6$*$-%%sqrtG6#,&\"\"*
\"\"\"*$)%#u|irG\"\"#F.!\"\"F.F1" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 45 "p3 := changevar(u = 3*sin(theta), p2, theta);" }}
{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#p3G-%$IntG6$,$*&-%%sqrtG6#,&\"\"*\"
\"\"*&F.F/)-%$sinG6#%&thetaG\"\"#F/!\"\"F/-%$cosGF4F/\"\"$F5" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 24 "assume(cos(theta) >= 0);" }}
}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 19 "p4 := simplify(p3);" }}
{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#p4G,$-%$IntG6$*$)-%$cosG6#%'theta|i
rG\"\"#\"\"\"F.\"\"*" }}}{EXCHG {PARA 256 "" 0 "" {TEXT -1 4 "16. " }
{XPPEDIT 18 0 "Int(arctan(x)/(1+x^2),x);" "6#-%$IntG6$*&-%'arctanG6#%
\"xG\"\"\",&F+F+*$F*\"\"#F+!\"\"F*" }{TEXT -1 0 "" }}}{EXCHG {PARA 0 "
> " 0 "" {MPLTEXT 1 0 34 "p1 := Int(arctan(x)/(1 + x^2), x);" }}{PARA 
11 "" 1 "" {XPPMATH 20 "6#>%#p1G-%$IntG6$*&-%'arctanG6#%\"xG\"\"\",&F-
F-*$)F,\"\"#F-F-!\"\"F," }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 38 "
p2 := changevar(u = arctan(x), p1, u);" }}{PARA 11 "" 1 "" {XPPMATH 
20 "6#>%#p2G-%$IntG6$%#u|irGF(" }}}{EXCHG {PARA 256 "" 0 "" {TEXT -1 
4 "17. " }{XPPEDIT 18 0 "Int(x^4/sqrt(x^10-4),x);" "6#-%$IntG6$*&%\"xG
\"\"%-%%sqrtG6#,&*$F'\"#5\"\"\"F(!\"\"F0F'" }{TEXT -1 0 "" }}}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 35 "p1 := Int(x^4 / sqrt(x^10 - 4), x);
" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#p1G-%$IntG6$*&%\"xG\"\"%,&*$)F)
\"#5\"\"\"F/F*!\"\"#F0\"\"#F)" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 
0 32 "p2 := changevar(u = x^5, p1, u);" }}{PARA 11 "" 1 "" {XPPMATH 
20 "6#>%#p2G-%$IntG6$,$*&\"\"\"F**$-%%sqrtG6#,&*$)%#u|irG\"\"#F*F*\"\"
%!\"\"F*F5#F*\"\"&F2" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 19 "p3 \+
:= simplify(p2);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#p3G,$-%$IntG6$*
&\"\"\"F**$-%%sqrtG6#,&*$)%#u|irG\"\"#F*F*\"\"%!\"\"F*F5F2#F*\"\"&" }}
}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 41 "p4 := changevar(u = 2*sec(p
hi), p3, phi);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#p4G,$-%$IntG6$,$*
(,&*$)-%$secG6#%$phiG\"\"#\"\"\"\"\"%F4!\"\"#F5F2F.F3-%$tanGF0F3F2F1#F
3\"\"&" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 21 "assume(tan(phi) >
 0);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 19 "p5 := simplify(p4);
" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#p5G,$-%$IntG6$*(-%$cosG6#%%phi|
irG!\"\",&\"\"\"F0*$)F*\"\"#F0F.#F.F3-%$sinGF,F0F-#F0\"\"&" }}}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 38 "p6 := subs(1/cos(phi) = sec(phi), p
5);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#p6G,$-%$IntG6$*(-%$secG6#%%p
hi|irG\"\"\",&F.F.*$)-%$cosGF,\"\"#F.!\"\"#F5F4-%$sinGF,F.F-#F.\"\"&" 
}}}{EXCHG {PARA 256 "" 0 "" {TEXT -1 4 "18. " }{XPPEDIT 18 0 "Int((x+2
)/sqrt(x^2+4*x+6),x);" "6#-%$IntG6$*&,&%\"xG\"\"\"\"\"#F)F)-%%sqrtG6#,
(*$F(F*F)*&\"\"%F)F(F)F)\"\"'F)!\"\"F(" }{TEXT -1 0 "" }}}{EXCHG 
{PARA 0 "" 0 "" {TEXT -1 130 "(This problem and the next two are relat
ed; we will use new variables so that we can refer back to this answer
 without confusion.)" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 40 "q1 \+
:= Int((x+2)/sqrt(x^2 + 4*x + 6), x);" }}{PARA 11 "" 1 "" {XPPMATH 20 
"6#>%#q1G-%$IntG6$*&,&%\"xG\"\"\"\"\"#F+F+,(*$)F*F,F+F+*&\"\"%F+F*F+F+
\"\"'F+#!\"\"F,F*" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 42 "q2 := \+
changevar(u = x^2 + 4*x + 6, q1, u);" }}{PARA 11 "" 1 "" {XPPMATH 20 "
6#>%#q2G-%$IntG6$,$*&\"\"\"F**$-%%sqrtG6#%#u|irGF*!\"\"#F*\"\"#F/" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 19 "q3 := simplify(q2);" }}
{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#q3G,$-%$IntG6$*&\"\"\"F**$-%%sqrtG6
#%#u|irGF*!\"\"F/#F*\"\"#" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 
16 "q4 := value(q3);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#q4G*$-%%sqr
tG6#%#u|irG\"\"\"" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 34 "q5 := \+
subs(u = x^2 + 4*x + 6, q4);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#q5G
*$-%%sqrtG6#,(*$)%\"xG\"\"#\"\"\"F.*&\"\"%F.F,F.F.\"\"'F.F." }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 12 "q1 = q5 + C;" }}{PARA 11 "" 
1 "" {XPPMATH 20 "6#/-%$IntG6$*&,&%\"xG\"\"\"\"\"#F*F*,(*$)F)F+F*F**&
\"\"%F*F)F*F*\"\"'F*#!\"\"F+F),&*$-%%sqrtG6#F,F*F*%\"CGF*" }}}{EXCHG 
{PARA 256 "" 0 "" {TEXT -1 4 "19. " }{XPPEDIT 18 0 "Int(2/sqrt(x^2+4*x
+6),x);" "6#-%$IntG6$*&\"\"#\"\"\"-%%sqrtG6#,(*$%\"xGF'F(*&\"\"%F(F.F(
F(\"\"'F(!\"\"F." }{TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 
1 0 36 "r1 := Int(2/sqrt(x^2 + 4*x + 6), x);" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#>%#r1G-%$IntG6$,$*&\"\"\"F**$-%%sqrtG6#,(*$)%\"xG\"\"#F
*F**&\"\"%F*F2F*F*\"\"'F*F*!\"\"F3F2" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 19 "r2 := simplify(r1);" }}{PARA 11 "" 1 "" {XPPMATH 20 "
6#>%#r2G,$-%$IntG6$*&\"\"\"F**$-%%sqrtG6#,(*$)%\"xG\"\"#F*F**&\"\"%F*F
2F*F*\"\"'F*F*!\"\"F2F3" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 36 "
cs := completesquare(x^2 + 4*x + 6);" }}{PARA 11 "" 1 "" {XPPMATH 20 "
6#>%#csG,&*$),&%\"xG\"\"\"\"\"#F*F+F*F*F+F*" }}}{EXCHG {PARA 0 "> " 0 
"" {MPLTEXT 1 0 35 "r3 := subs(x^2 + 4*x + 6 = cs, r2);" }}{PARA 11 "
" 1 "" {XPPMATH 20 "6#>%#r3G,$-%$IntG6$*&\"\"\"F**$-%%sqrtG6#,&*$),&%
\"xGF*\"\"#F*F4F*F*F4F*F*!\"\"F3F4" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 45 "r4 := changevar(x+2 = sqrt(2)*tan(u), r3, u);" }}
{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#r4G,$-%$IntG6$*(\"\"##\"\"\"F*,&F,F
,*$)-%$tanG6#%#u|irGF*F,F,F,,&F*F,*&F*F,F/F,F,#!\"\"F*F3F*" }}}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 19 "r5 := simplify(r4);" }}{PARA 11 "" 
1 "" {XPPMATH 20 "6#>%#r5G,$-%$IntG6$*$-%%sqrtG6#,&\"\"\"F.*$)-%$tanG6
#%#u|irG\"\"#F.F.F.F4F5" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 19 "
assume(sec(u) > 0);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 40 "r6 :
= subs(1 + tan(u)^2 = sec(u)^2, r5);" }}{PARA 11 "" 1 "" {XPPMATH 20 "
6#>%#r6G,$-%$IntG6$*$-%%sqrtG6#,&\"\"\"F.*$)-%$tanG6#%#u|irG\"\"#F.F.F
.F4F5" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 26 "r7 := simplify(r6,
 power);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#r7G,$-%$IntG6$*$-%%sqrt
G6#,&\"\"\"F.*$)-%$tanG6#%#u|irG\"\"#F.F.F.F4F5" }}}{EXCHG {PARA 0 "> \+
" 0 "" {MPLTEXT 1 0 16 "r8 := value(r7);" }}{PARA 11 "" 1 "" {XPPMATH 
20 "6#>%#r8G,$-%(arcsinhG6#-%$tanG6#%#u|irG\"\"#" }}}{EXCHG {PARA 0 ">
 " 0 "" {MPLTEXT 1 0 73 "r9 := subs(tan(u) = (x+2)/sqrt(2), sec(u) = s
qrt((x^2 + 4*x + 6)/2), r8);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#r9G
,$-%(arcsinhG6#-%$tanG6#%#u|irG\"\"#" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 20 "r10 := simplify(r9);" }}{PARA 11 "" 1 "" {XPPMATH 20 
"6#>%$r10G,$-%(arcsinhG6#-%$tanG6#%#u|irG\"\"#" }}}{EXCHG {PARA 0 "> \+
" 0 "" {MPLTEXT 1 0 13 "r1 = r10 + C;" }}{PARA 11 "" 1 "" {XPPMATH 20 
"6#/-%$IntG6$,$*&\"\"\"F)*$-%%sqrtG6#,(*$)%\"xG\"\"#F)F)*&\"\"%F)F1F)F
)\"\"'F)F)!\"\"F2F1,&-%(arcsinhG6#-%$tanG6#%#u|irGF2%\"CGF)" }}}
{EXCHG {PARA 256 "" 0 "" {TEXT -1 4 "20. " }{XPPEDIT 18 0 "Int(x/sqrt(
x^2+4*x+6),x);" "6#-%$IntG6$*&%\"xG\"\"\"-%%sqrtG6#,(*$F'\"\"#F(*&\"\"
%F(F'F(F(\"\"'F(!\"\"F'" }{TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 38 "s1 := Int(x / sqrt(x^2 + 4*x + 6), x);" }}{PARA 11 "
" 1 "" {XPPMATH 20 "6#>%#s1G-%$IntG6$*&%\"xG\"\"\",(*$)F)\"\"#F*F**&\"
\"%F*F)F*F*\"\"'F*#!\"\"F.F)" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 77 "O
bserve that this anti-derivative is just the difference of the previou
s two:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 18 "s1 = q5 - r10 + C
;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%$IntG6$*&%\"xG\"\"\",(*$)F(\"
\"#F)F)*&\"\"%F)F(F)F)\"\"'F)#!\"\"F-F(,(*$-%%sqrtG6#F*F)F)*&F-F)-%(ar
csinhG6#-%$tanG6#%#u|irGF)F2%\"CGF)" }}}}{MARK "0 1 0" 55 }{VIEWOPTS 
1 1 0 1 1 1803 1 1 1 1 }{PAGENUMBERS 0 1 2 33 1 1 }