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

Section 5.5 The Substitution Rule 417
Therefore
y 4
s2x 1 1 dx − y 9 1
2 su du
0 1
− 1 2 ? 2 3 u 3y2 g 1
9
− 1 3 s93y2 2 1 3y2 d − 26
3
Observe that when using (6) we do not return to the variable x after integrating. We
simply evaluate the expression in u between the appropriate values of u.
n
The integral given in Example 8 is an
abbreviation for
y 2 1
1 s3 2 5xd dx 2
Example 8 Evaluate y 2
1
dx
s3 2 5xd 2 .
SOLUtion Let u − 3 2 5x. Then du − 25 dx, so dx − 2 1 5 du. When x − 1, u − 22
and when x − 2, u − 27. Thus
y 2
1
dx
s3 2 5xd 2
− 2 1 5 y27 22
du
u 2
27
− 2 1 F2 1 − 1
5 uG22
27
5uG22
− 1 5
S2 1 7 1 1 2D − 1 14
n
Since the function f sxd − sln xdyx
in Example 9 is positive for x . 1,
the integral represents the area of the
shaded region in Figure 2.
y
0.5
y= ln x
x
Example 9 Calculate y e
1
ln x
x
dx.
SOLUtion We let u − ln x because its differential du − dxyx occurs in the integral.
When x − 1, u − ln 1 − 0; when x − e, u − ln e − 1. Thus
y e
1
ln x
x
dx − y 1
u du − u 2
0
1
2G0
− 1 2
n
0
FIGURE 2
1 e
x
Symmetry
The next theorem uses the Substitution Rule for Definite Integrals (6) to simplify the
calculation of integrals of functions that possess symmetry properties.
7 Integrals of Symmetric Functions Suppose f is continuous on f2a, ag.
(a) If f is even f f s2xd − f sxdg, then y a f sxd dx − 2 2a ya f sxd dx.
0
(b) If f is odd f f s2xd − 2f sxdg, then y a 2a
f sxd dx − 0.
Proof We split the integral in two:
8
y a 2a
f sxd dx − y 0 f sxd dx 1 y a
2a
0
f sxd dx − 2y 2a
0
f sxd dx 1 y a
f sxd dx
0
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.