James Stewart-Calculus_ Early Transcendentals-Cengage Learning (2015)

414 Chapter 5 Integrals
Check the answer by differentiating it.
x 3 dx − 1 4 du and the Substitution Rule, we have
y x 3 cossx 4 1 2d dx − y cos u ? 1 4 du − 1 4 y cos u du
− 1 4 sin u 1 C
− 1 4 sinsx 4 1 2d 1 C
Notice that at the final stage we had to return to the original variable x.
n
The idea behind the Substitution Rule is to replace a relatively complicated integral
by a simpler integral. This is accomplished by changing from the original variable x
to a new variable u that is a function of x. Thus in Example 1 we replaced the integral
y x 3 cossx 4 1 2d dx by the simpler integral 1 4 y cos u du.
The main challenge in using the Substitution Rule is to think of an appropriate substitution.
You should try to choose u to be some function in the integrand whose differential
also occurs (except for a constant factor). This was the case in Example 1. If that is not
pos sible, try choosing u to be some complicated part of the integrand (perhaps the inner
function in a composite function). Finding the right substitution is a bit of an art. It’s not
unusual to guess wrong; if your first guess doesn’t work, try another substitution.
Example 2 Evaluate y s2x 1 1 dx.
SOLUTION 1 Let u − 2x 1 1. Then du − 2 dx, so dx − 1 2 du. Thus the Substitution
Rule gives
y s2x 1 1 dx − y su
? 1 2 du − 1 2 y u 1y2 du
− 1 2 ? u 3y2
3y2 1 C − 1 3 u 3y2 1 C
− 1 3 s2x 1 1d3y2 1 C
SOLUTION 2 Another possible substitution is u − s2x 1 1. Then
dx
du −
s2x 1 1
so
dx − s2x 1 1 du − u du
(Or observe that u 2 − 2x 1 1, so 2u du − 2 dx.) Therefore
Example 3 Find y
y s2x 1 1 dx − y u ? u du − y u 2 du
− u 3
3 1 C − 1 3 s2x 1 1d3y2 1 C n
x
dx.
s1 2 4x 2
SOLUTION Let u − 1 2 4x 2 . Then du − 28x dx, so x dx − 2 1 8 du and
y
x
s1 2 4x 2 dx − 2 1 8y
1
su
du − 2 1 8y u 21y2 du
− 2 1 8 (2su ) 1 C − 2 1 4 s1 2 4x 2 1 C
n
Copyright 2016 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part. Due to electronic rights, some third party content may be suppressed from the eBook and/or eChapter(s).
Editorial review has deemed that any suppressed content does not materially affect the overall learning experience. Cengage Learning reserves the right to remove additional content at any time if subsequent rights restrictions require it.