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Chapter 34
34–1
◆
Methods of Integration
Integrals of Exponential and Logarithmic Functions
Integral of eu du
Since the derivative of eu is
d (eu )
du
dx
e u
dx
or d(eu) eu du, then integrating gives
eu du 3
d (eu) eu C
3
or the following:
eu du eu C
Rule 8
3
Example 1: Integrate 1e6x dx.
Solution: To match the form 1eu du, let
◆◆◆
u 6x
du 6 dx
We insert a factor of 6 and compensate with 16 ; then we use Rule 8.
1
1
e6x dx e6x (6 dx) e6x C
63
6
3
◆◆◆
We now do a definite integral. Simply substitute the limits, as before.
3
◆◆◆
Example 2: Integrate
6e
3x
dx.
0 3x
3
Solution: Since the derivative of 3x is 32 (3x)12 we insert a factor of 32 and compensate.
3
3x
3
6e
dx 6
0
3
3x
0
3
e
3x
p
(3x)12 dx
2
6 q
e
3 3
0
3
3x
3x
4e
[
3
2
(3x)12 dx
]
3
0
4e3 4e0 76.3
◆◆◆
Integral of bu du
The derivative of buln b is
d p bu q
dx ln b
1
ln b
du
dx
du
dx
(b ) (ln b) b u
u
or, in differential form, d(buln b) bu du. Thus the integral of bu du is as follows:
Rule 9
bu
bu du C (b 0, b 1)
ln b
3
Section 34–1
◆◆◆
◆
995
Integrals of Exponential and Logarithmic Functions
Example 3: Integrate
2
3
3xa2x dx.
Solution:
3xa2x dx 3 a2x x dx
3
3
1
2
3 p q a2x (4x dx)
4 3
2
3a2x
C
4 ln a
2
2
◆◆◆
Integral of ln u
To integrate the natural logarithm of a function, we use Rule 43, which we give here without
proof.
ln u du u (ln u 1) C
Rule 43
3
Example 4: Integrate 1x ln(3x2) dx.
Solution: We put our integral into the form of Rule 43 and integrate.
◆◆◆
p
q
p
q
x ln (3x2) dx 16
3
ln (3x2) (6x dx)
3
16 (3x2) (ln 3x2 1) C
12 x2 (ln 3x2 1) C
◆◆◆
Integral of log u
To integrate the common logarithm of a function, we first convert it to a natural logarithm using
Eq. 195.
ln N
ln N
log N ln 10 2.3026
195
4
log (3x 7) dx.
3
3
Solution: We convert the common log to a natural log and apply Rule 43.
◆◆◆
Example 5: Integrate
4
3
3
4
log (3x 7) dx 3
3
ln (3x 7)
dx
ln 10
4
1
ln (3x 7) (3 dx)
3 ln 10 3
3
1
4
(3x 7) ln (3x 7) 13
3 ln 10
1
5 (ln 5 1) 2 (ln 2 1)
3 ln 10
0.52997
◆◆◆
The argument of the natural
logarithm function must be
positive. For ease of reading, the
absolute value signs have been
omitted in this section.
996
◆
Chapter 34
Methods of Integration
Exercise 1
◆
Integrals of Exponential and Logarithmic Functions
Integrate.
Exponential Functions
Each of these problems has
been contrived to match a table
entry. But what if you had one
that did not match? We’ll learn
how to deal with those later in
this chapter.
a5x dx
2.
3
57x dx
4.
3
a3y dy
6.
3
4ex dx
8.
3
1.
3
3.
3
5.
3
7.
3
9.
2
xex dx
3
10.
a9x dx
10x dx
2
xa3x dx
e2x dx
3
x2ex dx
3
2
11.
13.
4
2
xe3x dx
3
1
e
x
dx
x
14.
(ex 1)2 dx
16.
(x 3) ex 6x2 dx
2
3
1
15.
17.
18.
3
0
3
x2
e
dx
3 x2
(ex2 ex2)2
dx
3
4
19.
3
20.
3
(exa exa) dx
(exa exa)2 dx
Logarithmic Functions
ln 3x dx
21.
3
22.
3
ln 7x dx
2
As usual, round all approximate
answers in this chapter to at
least three significant digits.
23.
3
1
24.
3
x ln x2 dx
log (5x 3) dx
4
25.
x log (x2 1) dx
3
2
3
26.
3
1
et dt
3
3
3
12.
x2 log (2 3x3) dx
xex dx
2
3
2
Section 34–2
◆
Integrals of the Trigonometric Functions
27. Find the area under the curve y e2x from x 1 to 3.
28. The first-quadrant area bounded by the catenary y 12 (ex ex) from x 0 to 1 is rotated
about the x axis. Find the volume generated.
29. The first-quadrant area bounded by y ex and x 1 is rotated about the line x 1. Find
the volume generated.
30. Find the length of the catenary y (a2)(exa exa) from x 0 to 6. Use a 3.
31. The curve y ex is rotated about the x axis. Find the area of the surface generated, from
x 0 to 100.
_
32. Find the horizontal distance x to the centroid of the area formed by the curve y 12 (ex ex), the coordinate axes, and the line x 1.
_
33. Find the vertical distance y to the centroid of the area formed by the curve y ex between
x 0 and 1.
34. A volume of revolution is formed by rotating the curve y ex between x 0 and 1 about
the x axis. Find the distance from the origin to the centroid.
35. Find the moment of inertia about the x and y axes for the area bounded by the curve y ex,
the line x 1, and the coordinate axes.
34–2 Integrals of the Trigonometric Functions
To our growing list of rules we add those for the six trigonometric functions. By Eq. 351,
d (cos u)
du
sin u dx
dx
or d(cos u) sin u du. Taking the integral of both sides gives
sin u du cos u C
3
The integrals of the other trigonometric functions are found in the same way. We thus get the
following rules:
Rule 10
Rule 11
Rule 12
Rule 13
Rule 14
Rule 15
sin u du cos u C
3
cos u du sin u C
3
tan u du ln cos u C
3
cot u du ln sin u C
3
sec u du ln sec u tan u C
3
csc u du ln csc u cot u C
3
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