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{SECT 0 {EXCHG {PARA 18 "" 0 "" {TEXT -1 11 "Calculus II" }}{PARA 257 
"" 0 "" {TEXT -1 38 "Lesson 8:  Integration by Substitution" }}}{SECT 
0 {PARA 3 "" 0 "" {TEXT -1 12 "Introduction" }}{EXCHG {PARA 0 "" 0 "" 
{TEXT -1 140 "We are about to study what is commonly called \"Techniqu
es of Integration\".  This is really a misnomer: what we will learn ar
e techniques of " }{TEXT 256 20 "anti-differentiation" }{TEXT -1 313 "
---the material of this chapter has nothing to do with partitioning, a
pproximating and summing.  Of course, the Fundamental Theorem of Calcu
lus tells you that if you can find an anti-derivative for a function t
hen you can integrate that function, which is why many people---and te
xtbooks---confuse the two topics." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}
{PARA 0 "" 0 "" {TEXT -1 469 "As we study this area, you may get the i
mpression that it's all just a bunch of tricks.  In a sense, that is t
rue.  Differentiation is a very systematic process: learn the product,
 quotient and chain rules, and a few basic derivatives, and you can di
fferentiate just about any function.  Anti-differentiation, on the oth
er hand, is far less automatic.  Consider the following examples.  (Ap
art from looking at the answers, notice two things: the use of the une
valuated " }{TEXT 272 3 "Int" }{TEXT -1 79 " command to print out the \+
integral which is being evaluated, and the fact that " }{TEXT 285 5 "M
aple" }{TEXT -1 93 " does not add the arbitrary constant to an anti-de
rivative, so we have to put it in by hand.)" }}}{EXCHG {PARA 0 "> " 0 
"" {MPLTEXT 1 0 52 "Int(x*sqrt(1 + x^2),x) = int(x*sqrt(1 + x^2),x) + \+
C;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%$IntG6$*&%\"xG\"\"\"-%%sqrtG6
#,&F)F)*$)F(\"\"#F)F)F)F(,&*$)F-#\"\"$F0F)#F)F5%\"CGF)" }}}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 56 "Int(x^2*sqrt(1 + x^2),x) = int(x^2*
sqrt(1 + x^2),x) + C;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%$IntG6$*&)
%\"xG\"\"#\"\"\"-%%sqrtG6#,&F+F+*$F(F+F+F+F),**&F)F+)F/#\"\"$F*F+#F+\"
\"%*&#F+\"\")F+*&F)F+F,F+F+!\"\"*&#F+F:F+-%(arcsinhG6#F)F+F<%\"CGF+" }
}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 56 "Int(x^3*sqrt(1 + x^2),x) =
 int(x^3*sqrt(1 + x^2),x) + C;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%$
IntG6$*&)%\"xG\"\"$\"\"\"-%%sqrtG6#,&F+F+*$)F)\"\"#F+F+F+F),(*&F1F+)F/
#F*F2F+#F+\"\"&*&#F2\"#:F+*$F5F+F+!\"\"%\"CGF+" }}}{EXCHG {PARA 0 "" 
0 "" {TEXT -1 467 "The first and third examples are not too surprising
: at least the answer is the same type of function as the integrand.  \+
It is not easy to see why the second answer should have an inverse hyp
erbolic function in it.  Perhaps this suggests why anti-differentiatio
n can be so frustrating: it is often impossible to guess the type of f
unctions that will appear in an anti-derivative, and changing the inte
grand slightly can produce a completely different anti-derivative." }}
{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 80 "Even find
ing an anti-derivative may not be very useful.  Look at these examples
:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 40 "Int(exp(-x^2),x) = int
(exp(-x^2),x) + C;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%$IntG6$-%$exp
G6#,$*$)%\"xG\"\"#\"\"\"!\"\"F-,&*&-%%sqrtG6#%#PiGF/-%$erfG6#F-F/#F/F.
%\"CGF/" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 44 "Int(sqrt(1 + x^3
),x) = int(sqrt(1 + x^3),x);" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#/-%$In
tG6$*$-%%sqrtG6#,&\"\"\"F,*$)%\"xG\"\"$F,F,F,F/,&*&F/F,F(F,#\"\"#\"\"&
*&*.#\"\"'F5F,,&#F0F4F,*&^##!\"\"F4F,-F)6#F0F,F,F,-F)6#*&,&F/F,F,F,F,F
:F?F,-F)6#*&,(F/F,#F,F4F?F<F,F,,&#!\"$F4F,F<F,F?F,-F)6#*&,(F/F,#F,F4F?
*&^##F,F4F,F@F,F,F,,&FLF,FSF,F?F,-%*EllipticFG6$*$FBF,*$-F)6#*&FVF,FKF
?F,F,F,*$-F)6#F+F,F?F," }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }
{TEXT 257 5 "Maple" }{TEXT -1 117 " can find these anti-derivatives, b
ut they are expressed in terms of functions you have probably never he
ard of, the " }{TEXT 258 14 "error function" }{TEXT -1 1 " " }{TEXT 
259 3 "erf" }{TEXT -1 9 " and the " }{TEXT 260 17 "elliptic function" 
}{TEXT -1 1 " " }{TEXT 261 10 " EllipticF" }{TEXT -1 177 ".  Unless yo
u know the definitions of these functions, the answers are no use to y
ou.  (They are not much use even if you do know their definitions, bec
ause these functions are " }{TEXT 262 7 "defined" }{TEXT -1 185 " in t
erms of the respective anti-derivatives.)  These examples show that it
 is not enough to have an anti-derivative: to be useful, the anti-deri
vative must be written in terms of known" }}{PARA 0 "" 0 "" {TEXT -1 
114 "functions.  Traditionally, we consider an anti-derivative to be u
seful if it is written in terms of the so-called " }{TEXT 263 21 "elem
entary functions:" }{TEXT -1 155 " polynomials, rational functions, ex
ponentials, trigonometric functions and their inverses.  (Note that th
is includes logarithms and hyperbolic functions.)" }}{PARA 0 "" 0 "" 
{TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 42 "We have to face one diffi
culty when using " }{TEXT 264 5 "Maple" }{TEXT -1 117 " to study techn
iques of anti-differentiation: it is just too powerful.  Any problem i
n this chapter can be solved in " }{TEXT 265 5 "Maple" }{TEXT -1 10 " \+
with one " }{TEXT 266 3 "int" }{TEXT -1 108 " command, so we will have
 to deliberately tie our hands.  We will allow ourselves to check answ
ers by using " }{TEXT 268 3 "int" }{TEXT -1 64 " if necessary (though \+
it is probably better to check them using " }{TEXT 269 4 "diff" }
{TEXT -1 35 "), but we will use the unevaluated " }{TEXT 270 3 "Int" }
{TEXT -1 191 " command while we are working out the answer.  The game \+
is to use one or more of the methods we will study to convert a given \+
anti-derivative problem into one we can recognise and evaluate.  " }}
{PARA 0 "" 0 "" {TEXT 267 0 "" }}{PARA 0 "" 0 "" {TEXT -1 214 "While t
here are many tricks involved in anti-differentiation, there are 2 mai
n methods.   In the rest of this worksheet, we will look at some simpl
e examples of these methods, and see how we will implement them in " }
{TEXT 271 5 "Maple" }{TEXT -1 1 "." }}}}{SECT 0 {PARA 3 "" 0 "" {TEXT 
-1 36 "Anti-differentiation by Substitution" }}{EXCHG {PARA 0 "> " 0 "
" {MPLTEXT 1 0 8 "restart;" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 50 "If \+
you were asked to evaluate the anti-derivative " }{XPPEDIT 18 0 "Int(x
*sqrt(1+x^2),x);" "6#-%$IntG6$*&%\"xG\"\"\"-%%sqrtG6#,&F(F(*$F'\"\"#F(
F(F'" }{TEXT -1 43 " by hand, you should think of substituting " }
{XPPEDIT 18 0 "u = 1+x^2;" "6#/%\"uG,&\"\"\"F&*$%\"xG\"\"#F&" }{TEXT 
-1 79 ".  (You should probably work through this example by hand befor
e solving it in " }{TEXT 273 5 "Maple" }{TEXT -1 2 ".)" }}{PARA 0 "" 
0 "" {TEXT -1 11 "We can get " }{TEXT 274 5 "Maple" }{TEXT -1 53 " to \+
work through the substitution by using a command " }{TEXT 275 9 "chang
evar" }{TEXT -1 14 ".   Like many " }{TEXT 276 5 "Maple" }{TEXT -1 97 
" commands, this one is not loaded automatically when the program star
ts.  It is contained in the " }{TEXT 277 7 "student" }{TEXT -1 63 " pa
ckage, and so this package must be loaded before we can use " }{TEXT 
278 9 "changevar" }{TEXT -1 7 ".  The " }{TEXT 279 5 "Maple" }{TEXT 
-1 33 " command that loads a package is " }{TEXT 280 4 "with" }{TEXT 
-1 2 ". " }}}{EXCHG {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%*changevarG%/completesqua
reG%)distanceG%'equateG%*integrandG%*interceptG%)intpartsG%(leftboxG%(
leftsumG%)makeprocG%*middleboxG%*middlesumG%)midpointG%(powsubsG%)righ
tboxG%)rightsumG%,showtangentG%(simpsonG%&slopeG%(summandG%*trapezoidG
" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 6 "(When " }{TEXT 281 5 "Maple" }
{TEXT -1 151 " loads a package, the commands contained in the package \+
are listed.  If you are familiar with the package, you can suppress th
is listing by ending the " }{TEXT 282 4 "with" }{TEXT -1 161 " command
 with a colon.)  Once a package has been loaded, you can use any comma
nd in the package in all the usual ways.  In particular, you can get h
elp for them." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" 
{TEXT -1 38 "Now follow through the computation of " }{XPPEDIT 18 0 "I
nt(x*sqrt(1+x^2),x);" "6#-%$IntG6$*&%\"xG\"\"\"-%%sqrtG6#,&F(F(*$F'\"
\"#F(F(F'" }{TEXT -1 7 " using " }{TEXT 283 9 "changevar" }{TEXT -1 
33 ".  We first define an expression " }{TEXT 284 2 "p1" }{TEXT -1 87 
" to be the (unevaluated) integral, then we make the suggested substit
ution. The use of " }{TEXT 286 8 "simplify" }{TEXT -1 119 " in the thi
rd line is optional: in any given example it may or may not help.  On \+
the other hand, the third argument in " }{TEXT 287 9 "changevar" }
{TEXT -1 55 " is necessary: there are two variables in the problem (" 
}{XPPEDIT 18 0 "x;" "6#%\"xG" }{TEXT -1 5 " and " }{XPPEDIT 18 0 "u;" 
"6#%\"uG" }{TEXT -1 21 "), and you must tell " }{TEXT 288 5 "Maple" }
{TEXT -1 56 " which of them is to be the new variable of integration.
" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 29 "p1 := Int(x*sqrt(1 + x^
2),x);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#p1G-%$IntG6$*&%\"xG\"\"\"
-%%sqrtG6#,&F*F**$)F)\"\"#F*F*F*F)" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 33 "p2 := changevar(u=1 + x^2, p1,u);" }}{PARA 11 "" 1 "
" {XPPMATH 20 "6#>%#p2G-%$IntG6$,$*$-%%sqrtG6#%\"uG\"\"\"#F.\"\"#F-" }
}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 19 "p3 := simplify(p2);" }}
{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#p3G,$-%$IntG6$*$-%%sqrtG6#%\"uG\"\"
\"F-#F.\"\"#" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 61 " The last express
ion is a known integral---a simple power of " }{XPPEDIT 18 0 "u;" "6#%
\"uG" }{TEXT -1 63 "---and so we will allow ourselves to evaluate it. \+
 The command " }{TEXT 289 5 "value" }{TEXT -1 41 " will evaluate an un
evaluated expression." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 16 "p4
 := value(p3);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#p4G,$*$)%\"uG#\"
\"$\"\"#\"\"\"#F,F*" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 57 "Finally, o
f course, we must write the answer in terms of " }{XPPEDIT 18 0 "x;" "
6#%\"xG" }{TEXT -1 34 ", using the substitution equation " }{XPPEDIT 
18 0 "u = x^2+1.;" "6#/%\"uG,&*$%\"xG\"\"#\"\"\"$F)\"\"!F)" }{TEXT -1 
120 "  We first find one anti-derivative, then the general one.  This \+
makes it easier to check the answer by differentiation." }}}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 25 "p5 := subs(u=x^2 + 1,p4);" }}{PARA 
11 "" 1 "" {XPPMATH 20 "6#>%#p5G,$*$),&\"\"\"F)*$)%\"xG\"\"#F)F)#\"\"$
F-F)#F)F/" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 12 "p1 = p5 + C;" 
}}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%$IntG6$*&%\"xG\"\"\"-%%sqrtG6#,&F
)F)*$)F(\"\"#F)F)F)F(,&*$)F-#\"\"$F0F)#F)F5%\"CGF)" }}}{EXCHG {PARA 0 
"" 0 "" {TEXT -1 31 "You can check this answer with " }{TEXT 290 3 "in
t" }{TEXT -1 119 ", but you should get into the habit of checking anti
-derivatives by differentiating them, so let's do that.  Note that " }
{TEXT 291 2 "p5" }{TEXT -1 41 " is an expression, so we can use it in \+
a " }{TEXT 292 4 "diff" }{TEXT -1 9 " command:" }}}{EXCHG {PARA 0 "> \+
" 0 "" {MPLTEXT 1 0 11 "diff(p5,x);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6
#*&%\"xG\"\"\"-%%sqrtG6#,&F%F%*$)F$\"\"#F%F%F%" }}}{EXCHG {PARA 0 "" 
0 "" {TEXT -1 18 "The derivative of " }{TEXT 293 2 "p5" }{TEXT -1 69 "
 is what it should be, so we have found the correct anti-derivative. \+
" }}}{SECT 0 {PARA 4 "" 0 "" {TEXT -1 10 "Question 1" }}{EXCHG {PARA 
0 "" 0 "" {TEXT -1 258 "Use the method described above to find the fol
lowing anti-derivatives, then check them by differentiation.  (All of \+
these are taken from the Chapter 5 Review Exercises in Stewart, and yo
u should do them all---and others---with pencil and paper for homework
.)" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 4 "(a) \+
" }{XPPEDIT 18 0 "Int(x^2*(1+2*x^3)^3,x);" "6#-%$IntG6$*&%\"xG\"\"#,&
\"\"\"F**&F(F**$F'\"\"$F*F*F-F'" }{TEXT -1 2 "\n\n" }{TEXT 294 9 "Solu
tion." }{TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 34 "p1 \+
:= Int(x^2 * (1 + 2*x^3)^3, x);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#
p1G-%$IntG6$*&)%\"xG\"\"#\"\"\"),&F,F,*&F+F,)F*\"\"$F,F,F1F,F*" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 34 "p2 := changevar(u=1+2*x^3, p
1, u);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#p2G-%$IntG6$,$*$)%\"uG\"
\"$\"\"\"#F-\"\"'F+" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 19 "p3 :
= simplify(p2);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#p3G,$-%$IntG6$*$
)%\"uG\"\"$\"\"\"F+#F-\"\"'" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 
16 "p4 := value(p3);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#p4G,$*$)%\"
uG\"\"%\"\"\"#F*\"#C" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 26 "p5 \+
:= subs(u=1+2*x^3, p4);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#p5G,$*$)
,&\"\"\"F)*&\"\"#F))%\"xG\"\"$F)F)\"\"%F)#F)\"#C" }}}{EXCHG {PARA 0 ">
 " 0 "" {MPLTEXT 1 0 12 "p1 = p5 + C;" }}{PARA 11 "" 1 "" {XPPMATH 20 
"6#/-%$IntG6$*&)%\"xG\"\"#\"\"\"),&F+F+*&F*F+)F)\"\"$F+F+F0F+F),&*$)F-
\"\"%F+#F+\"#C%\"CGF+" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 6 "Check:" }
}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 11 "diff(p5,x);" }}{PARA 11 "
" 1 "" {XPPMATH 20 "6#*&)%\"xG\"\"#\"\"\"),&F'F'*&F&F')F%\"\"$F'F'F,F'
" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 100 "(By the way: I just got this
 one wrong--I mistyped the final substitution--so the check was useful
!)" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 4 "(b) " }{XPPEDIT 18 0 "Int(ex
p(sqrt(x))/sqrt(x),x);" "6#-%$IntG6$*&-%$expG6#-%%sqrtG6#%\"xG\"\"\"-F
+6#F-!\"\"F-" }{TEXT -1 2 "\n\n" }{TEXT 295 9 "Solution." }{TEXT -1 0 
"" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 34 "p1 := Int(exp(sqrt(x))
/sqrt(x),x);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#p1G-%$IntG6$*&-%$ex
pG6#*$-%%sqrtG6#%\"xG\"\"\"F1*$-F.6#F0F1!\"\"F0" }}}{EXCHG {PARA 0 "> \+
" 0 "" {MPLTEXT 1 0 32 "p2 := changevar(u=sqrt(x),p1,u);" }}{PARA 11 "
" 1 "" {XPPMATH 20 "6#>%#p2G-%$IntG6$,$-%$expG6#%\"uG\"\"#F," }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 19 "p3 := simplify(p2);" }}
{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#p3G,$-%$IntG6$-%$expG6#%\"uGF,\"\"#
" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 16 "p4 := value(p3);" }}
{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#p4G,$-%$expG6#%\"uG\"\"#" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 26 "p5 := subs(u=sqrt(x), p4);" 
}}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#p5G,$-%$expG6#*$-%%sqrtG6#%\"xG\"
\"\"\"\"#" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 12 "p1 = p5 + C;" 
}}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%$IntG6$*&-%$expG6#*$-%%sqrtG6#%\"
xG\"\"\"F0*$-F-6#F/F0!\"\"F/,&F(\"\"#%\"CGF0" }}}{EXCHG {PARA 0 "" 0 "
" {TEXT -1 6 "Check:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 11 "dif
f(p5,x);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#*&-%$expG6#*$-%%sqrtG6#%\"
xG\"\"\"F,*$-F)6#F+F,!\"\"" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 4 "(c) \+
" }{XPPEDIT 18 0 "Int(sec(theta)*tan(theta)/(1+sec(theta)),theta);" "6
#-%$IntG6$*(-%$secG6#%&thetaG\"\"\"-%$tanG6#F*F+,&F+F+-F(6#F*F+!\"\"F*
" }{TEXT -1 2 "\n\n" }{TEXT 296 9 "Solution." }}}{EXCHG {PARA 0 "> " 
0 "" {MPLTEXT 1 0 56 "p1 := Int(sec(theta)*tan(theta)/(1 + sec(theta))
,theta);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#p1G-%$IntG6$*&*&-%$secG
6#%&thetaG\"\"\"-%$tanGF,F.F.,&F.F.F*F.!\"\"F-" }}}{EXCHG {PARA 0 "> \+
" 0 "" {MPLTEXT 1 0 37 "p2 := changevar(u=1+sec(theta),p1,u);" }}
{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#p2G-%$IntG6$*&\"\"\"F)%\"uG!\"\"F*
" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 16 "p3 := value(p2);" }}
{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#p3G-%#lnG6#%\"uG" }}}{EXCHG {PARA 
0 "> " 0 "" {MPLTEXT 1 0 30 "p4 := subs(u=1+sec(theta),p3);" }}{PARA 
11 "" 1 "" {XPPMATH 20 "6#>%#p4G-%#lnG6#,&\"\"\"F)-%$secG6#%&thetaGF)
" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 12 "p1 = p4 + C;" }}{PARA 
11 "" 1 "" {XPPMATH 20 "6#/-%$IntG6$*&*&-%$secG6#%&thetaG\"\"\"-%$tanG
F+F-F-,&F-F-F)F-!\"\"F,,&-%#lnG6#F0F-%\"CGF-" }}}{EXCHG {PARA 0 "" 0 "
" {TEXT -1 6 "Check:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 15 "dif
f(p4,theta);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#*&*&-%$secG6#%&thetaG
\"\"\"-%$tanGF'F)F),&F)F)F%F)!\"\"" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 
-1 4 "(d) " }{XPPEDIT 18 0 "Int(x/sqrt(1-x^4),x);" "6#-%$IntG6$*&%\"xG
\"\"\"-%%sqrtG6#,&F(F(*$F'\"\"%!\"\"F/F'" }{TEXT -1 2 "\n\n" }{TEXT 
297 9 "Solution." }{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
*F3F)" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 30 "p2 := changevar(u=
x^2, p1, u);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#p2G-%$IntG6$,$*&\"
\"\"F**$-%%sqrtG6#,&F*F**$)%\"uG\"\"#F*!\"\"F*F4#F*F3F2" }}}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 19 "p3 := simplify(p2);" }}{PARA 11 "" 
1 "" {XPPMATH 20 "6#>%#p3G,$-%$IntG6$*&\"\"\"F**$-%%sqrtG6#,&F*F**$)%
\"uG\"\"#F*!\"\"F*F4F2#F*F3" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 
16 "p4 := value(p3);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#p4G,$-%'arc
sinG6#%\"uG#\"\"\"\"\"#" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 22 "
p5 := subs(u=x^2, p4);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#p5G,$-%'a
rcsinG6#*$)%\"xG\"\"#\"\"\"#F-F," }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 12 "p1 = p5 + C;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%$I
ntG6$*&%\"xG\"\"\"*$-%%sqrtG6#,&F)F)*$)F(\"\"%F)!\"\"F)F2F(,&-%'arcsin
G6#*$)F(\"\"#F)#F)F9%\"CGF)" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 6 "Che
ck:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 11 "diff(p5,x);" }}
{PARA 11 "" 1 "" {XPPMATH 20 "6#*&%\"xG\"\"\"*$-%%sqrtG6#,&F%F%*$)F$\"
\"%F%!\"\"F%F." }}}}}}{MARK "1 1 0 0" 5 }{VIEWOPTS 1 1 0 1 1 1803 1 1 
1 1 }{PAGENUMBERS 0 1 2 33 1 1 }