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Section 5.2
The Natural Logarithmic Function: Integration
• Use the Log Rule for Integration to integrate a rational function.
• Integrate trigonometric functions.
E X P L O R AT I O N
Integrating Rational Functions
Early in Chapter 4, you learned rules
that allowed you to integrate any
polynomial function. The Log Rule
presented in this section goes a long
way toward enabling you to integrate
rational functions. For instance, each
of the following functions can be
integrated with the Log Rule.
Log Rule for Integration
The differentiation rules
d
1
ln x dx
x
and
u
d
ln u dx
u
that you studied in the preceding section produce the following integration rule.
THEOREM 5.5
Log Rule for Integration
Let u be a differentiable function of x.
2
x
Example 1
1
4x 1
Example 2
x
x2 1
Example 3
3x 2 1
x3 x
Example 4(a)
x1
x 2 2x
Example 4(c)
1
3x 2
Example 4(d)
x2 x 1
x2 1
Example 5
2x
x 1 2
Example 6
There are still some rational functions
that cannot be integrated using the
Log Rule. Give examples of these
functions, and explain your reasoning.
1.
1
dx ln x C
x
2.
1
du ln u C
u
Because du u dx, the second formula can also be written as
u
dx ln u C.
u
EXAMPLE 1
Alternative form of Log Rule
Using the Log Rule for Integration
2
1
dx 2
dx
x
x
2 ln x C
lnx 2 C
Constant Multiple Rule
Log Rule for Integration
Property of logarithms
Because x 2 cannot be negative, the absolute value is unnecessary in the final form of
the antiderivative.
EXAMPLE 2
Find
Using the Log Rule with a Change of Variables
1
dx.
4x 1
Solution If you let u 4x 1, then du 4 dx.
1
1
dx 4x 1
4
1
4
1
4
1
4
1
4 dx
4x 1
1
du
u
Substitute: u 4x 1.
ln u C
Multiply and divide by 4.
ln 4x 1 C
Apply Log Rule.
Back-substitute.
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SECTION 5.2
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333
Example 3 uses the alternative form of the Log Rule. To apply this rule, look for
quotients in which the numerator is the derivative of the denominator.
EXAMPLE 3
Finding Area with the Log Rule
Find the area of the region bounded by the graph of
y
x2
x
1
the x-axis, and the line x 3.
y
y=
0.5
Solution From Figure 5.8, you can see that the area of the region is given by the
definite integral
x
x2 + 1
3
0.4
0
0.3
x
dx.
x2 1
If you let u x2 1, then u 2x. To apply the Log Rule, multiply and divide by 2
as shown.
0.2
3
0.1
0
x
1
2
3
x2
x
1
2x
dx dx
1
2 0 x2 1
3
1
lnx 2 1
2
0
1
ln 10 ln 1
2
1
ln 10
2
1.151
3
3
x
Area dx
2
x
1
0
The area of the region bounded by the graph
of y, the x-axis, and x 3 is 12 ln 10.
Figure 5.8
EXAMPLE 4
a.
b.
c.
d.
u
dx ln u C
u
ln 1 0
Recognizing Quotient Forms of the Log Rule
3x 2 1
dx ln x 3 x C
x3 x
sec2 x
dx ln tan x C
tan x
1 2x 2
x1
dx dx
2
x 2x
2 x 2 2x
1
ln x2 2x C
2
Multiply and divide by 2.
u tan x
u x 2 2x
1
1
3
dx dx
3x 2
3 3x 2
1
ln 3x 2 C
3
u x3 x
u 3x 2
With antiderivatives involving logarithms, it is easy to obtain forms that look
quite different but are still equivalent. For instance, which of the following are
equivalent to the antiderivative listed in Example 4(d)?
ln 3x 213 C,
1
2
ln x 3 C,
3
13 C
ln 3x 2
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Integrals to which the Log Rule can be applied often appear in disguised form.
For instance, if a rational function has a numerator of degree greater than or equal to
that of the denominator, division may reveal a form to which you can apply the Log
Rule. This is shown in Example 5.
EXAMPLE 5
Find
Using Long Division Before Integrating
x2 x 1
dx.
x2 1
Solution Begin by using long division to rewrite the integrand.
1
x2 1 ) x2 x 1
x2
1
x
x2 x 1
x2 1
1
x
x2 1
Now, you can integrate to obtain
x2 x 1
dx x2 1
x
dx
1
1
2x
dx dx
2
2 x 1
1
x lnx 2 1 C.
2
1
x2
Rewrite using long division.
Rewrite as two integrals.
Integrate.
Check this result by differentiating to obtain the original integrand.
The next example gives another instance in which the use of the Log Rule is
disguised. In this case, a change of variables helps you recognize the Log Rule.
EXAMPLE 6
Find
Change of Variables with the Log Rule
2x
dx.
x 12
Solution If you let u x 1, then du dx and x u 1.
TECHNOLOGY If you have
access to a computer algebra system,
use it to find the indefinite integrals in
Examples 5 and 6. How does the form
of the antiderivative that it gives you
compare with that given in Examples
5 and 6?
2u 1
du
u2
u
1
2
du
u2 u2
du
2
2 u2 du
u
u1
2 ln u 2
C
1
2
2 ln u C
u
2
2 ln x 1 C
x1
2x
dx x 12
Substitute.
Rewrite as two fractions.
Rewrite as two integrals.
Integrate.
Simplify.
Back-substitute.
Check this result by differentiating to obtain the original integrand.
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335
As you study the methods shown in Examples 5 and 6, be aware that both
methods involve rewriting a disguised integrand so that it fits one or more of the basic
integration formulas. Throughout the remaining sections of Chapter 5 and in Chapter
8, much time will be devoted to integration techniques. To master these techniques,
you must recognize the “form-fitting” nature of integration. In this sense, integration
is not nearly as straightforward as differentiation. Differentiation takes the form
“Here is the question; what is the answer?”
Integration is more like
“Here is the answer; what is the question?”
The following are guidelines you can use for integration.
Guidelines for Integration
STUDY TIP Keep in mind that you
can check your answer to an integration
problem by differentiating the answer.
For instance, in Example 7, the derivative
of y ln ln x C is y 1x ln x.
1. Learn a basic list of integration formulas. (Including those given in this
section, you now have 12 formulas: the Power Rule, the Log Rule, and ten
trigonometric rules. By the end of Section 5.7, this list will have expanded to
20 basic rules.)
2. Find an integration formula that resembles all or part of the integrand, and,
by trial and error, find a choice of u that will make the integrand conform to
the formula.
3. If you cannot find a u-substitution that works, try altering the integrand. You
might try a trigonometric identity, multiplication and division by the same
quantity, or addition and subtraction of the same quantity. Be creative.
4. If you have access to computer software that will find antiderivatives
symbolically, use it.
EXAMPLE 7
u-Substitution and the Log Rule
Solve the differential equation
1 .
dy
dx x ln x
Solution The solution can be written as an indefinite integral.
y
1
dx
x ln x
Because the integrand is a quotient whose denominator is raised to the first power, you
should try the Log Rule. There are three basic choices for u. The choices u x and
u x ln x fail to fit the uu form of the Log Rule. However, the third choice does fit.
Letting u ln x produces u 1x, and you obtain the following.
1
dx x ln x
1x
dx
ln x
u
dx
u
ln u C
ln ln x C
So, the solution is y lnln x C.
Divide numerator and denominator by x.
Substitute: u ln x.
Apply Log Rule.
Back-substitute.
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Integrals of Trigonometric Functions
In Section 4.1, you looked at six trigonometric integration rules—the six that correspond directly to differentiation rules. With the Log Rule, you can now complete the
set of basic trigonometric integration formulas.
Using a Trigonometric Identity
EXAMPLE 8
Find
tan x dx.
Solution This integral does not seem to fit any formulas on our basic list. However,
by using a trigonometric identity, you obtain
tan x dx sin x
dx.
cos x
Knowing that Dx cos x sin x, you can let u cos x and write
sin x
dx
cos x
u
dx
u
ln u C
ln cos x C.
tan x dx Trigonometric identity
Substitute: u cos x.
Apply Log Rule.
Back-substitute.
Example 8 uses a trigonometric identity to derive an integration rule for the
tangent function. The next example takes a rather unusual step (multiplying and
dividing by the same quantity) to derive an integration rule for the secant function.
Derivation of the Secant Formula
EXAMPLE 9
Find
sec x dx.
Solution Consider the following procedure.
sec x dx sec x
x tan x
dx
sec
sec x tan x sec 2 x sec x tan x
dx
sec x tan x
Letting u be the denominator of this quotient produces
u sec x tan x
u sec x tan x sec 2 x.
So, you can conclude that
sec 2 x sec x tan x
dx
sec x tan x
u
dx
u
ln u C
ln sec x tan x C.
sec x dx Rewrite integrand.
Substitute: u sec x tan x.
Apply Log Rule.
Back-substitute.
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SECTION 5.2
337
The Natural Logarithmic Function: Integration
With the results of Examples 8 and 9, you now have integration formulas for
sin x, cos x, tan x, and sec x. All six trigonometric rules are summarized below.
NOTE Using trigonometric identities
and properties of logarithms, you could
rewrite these six integration rules in
other forms. For instance, you could
write
csc u du ln csc u cot u C.
Integrals of the Six Basic Trigonometric Functions
sin u du cos u C
cos u du sin u C
tan u du ln cos u C
EXAMPLE 10
cot u du ln sin u C
sec u du ln sec u tan u C
(See Exercises 83–86.)
csc u du ln csc u cot u C
Integrating Trigonometric Functions
4
Evaluate
1 tan2 x dx.
0
Solution Using 1 tan 2 x sec2 x, you can write
4
1 tan2 x dx 0
4
sec 2 x dx
0
4
sec x ≥ 0 for 0 ≤ x ≤
sec x dx
0
.
4
4
0
ln sec x tan x
ln2 1 ln 1
0.881.
EXAMPLE 11
Finding an Average Value
4 .
Find the average value of f x tan x on the interval 0,
Solution
y
2
f (x) = tan x
π
4
Average value ≈ 0.441
1
Figure 5.9
4
1
tan x dx
4 0 0
4 4
tan x dx
0
4
4
ln cos x
0
2
4
ln
ln1
2
Average value x
Average value Simplify.
Integrate.
2
4
ln
2
0.441
The average value is about 0.441, as shown in Figure 5.9.
1
ba
b
a
f x dx
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Logarithmic, Exponential, and Other Transcendental Functions
Exercises for Section 5.2
In Exercises 1–24, find the indefinite integral.
1.
3.
5.
7.
9.
11.
13.
15.
17.
19.
21.
23.
5
dx
x
2.
1
dx
x1
4.
1
dx
3 2x
6.
x
dx
x2 1
8.
x2 4
dx
x
10.
x 2 2x 3
dx
x 3 3x 2 9x
12.
x 2 3x 2
dx
x1
14.
x 3 3x 2 5
dx
x3
16.
x4 x 4
dx
x2 2
18.
ln x2
dx
x
1
dx
x 1
2x
dx
x 1 2
20.
22.
24.
10
dx
x
See www.CalcChat.com for worked-out solutions to odd-numbered exercises.
In Exercises 37– 40, solve the differential equation. Use a
graphing utility to graph three solutions, one of which passes
through the given point.
1
dx
x5
37.
dy
3
, 1, 0
dx 2 x
1
dx
3x 2
39.
ds
tan 2, 0, 2
d
x2
dx
3 x3
40.
dr
sec2 t
,
dt tan t 1
x
dx
9 x 2
xx 2
dx
x 3 3x 2 4
41. Determine the function f if f x x 3 3x 2 4x 9
dx
x2 3
1
dx
x lnx3
1
dx
x231 x13
xx 2
dx
x 1 3
, 4
42. Determine the function f if f x Slope Fields In Exercises 43– 46, a differential equation, a
point, and a slope field are given. (a) Sketch two approximate
solutions of the differential equation on the slope field, one of
which passes through the given point. (b) Use integration to find
the particular solution of the differential equation and use a
graphing utility to graph the solution. Compare the result with
the sketches in part (a). To print an enlarged copy of the graph,
go to the website www.mathgraphs.com.
dy
1
, 0, 1
dx x 2
43.
dy ln x
, 1, 2
dx
x
44.
y
y
25.
27.
dx
1 2x
x
dx
x 3
26.
28.
31.
33.
34.
35.
36.
cos d
sin 30.
csc 2x dx
32.
cos t
dt
1 sin t
csc2 t
dt
cot t
sec x tan x
dx
sec x 1
sec t tan t dt
3
3
2
1
dx
1 3x
3 x
dx
3
x1
1
x
−2
tan 5 d
sec
x
−1
4
5
−2
−3
In Exercises 29–36, find the indefinite integral.
29.
4
2, f 2 3,
x 12
f2 0, x > 1.
In Exercises 25–28, find the indefinite integral by u-substitution.
(Hint: Let u be the denominator of the integrand.)
1
2
, f 1 1,
x2
f1 1, x > 0.
2x 2 7x 3
dx
x2
x 3 6x 20
dx
x5
dy
2x
, 0, 4
dx x2 9
38.
1
dy
1 , 1, 4
dx
x
45.
x
dx
2
−3
46.
dy
sec x, 0, 1
dx
y
y
4
4
3
2
1
x
−1
−2
−π
2
π
2
6
−4
x
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SECTION 5.2
In Exercises 47–54, evaluate the definite integral. Use a graphing
utility to verify your result.
4
47.
0
e
48.
50.
e
1
52.
57.
4
56.
58.
1
x
x1
dx
x1
−π
2
x
60.
63. F x 1
x2 4
, x 1, x 4, y 0
x
x2
dx
x1
72. y x4
, x 1, x 4, y 0
x
sin2 x cos2 x
dx
cos x
4
62. F x 0
x2
1
dt
t
64. Fx 1
(b) 6
1
dt
t
(c)
1
2
75.
1
4
76.
0
(d) 1.25
(c) 2
(d) 5
6
77.
78.
(e) 1
ln x dx
sec x dx
Writing About Concepts
2
x ln x
In Exercises 79–82, state the integration formula you would
use to perform the integration. Do not integrate.
79.
y
y
80.
4
4
81.
3
2
2
82.
x
4
8x
dx
x2 4
3
(e) 3
68. y 2
12
dx
x
3
4
x
−2
x
, x 0, x 2, y 0
6
2
Area In Exercises 67–70, find the area of the given region. Use
a graphing utility to verify your result.
67. y 73. y 2 sec
5
2x
, 0, 4
66. f x 2
x 1
(b) 7
x
Numerical Integration In Exercises 75–78, use the Trapezoidal
Rule and Simpson’s Rule to approximate the value of the
definite integral. Let n 4 and round your answer to four
decimal places. Use a graphing utility to verify your result.
tan t dt
65. f x sec x, 0, 1
(a) 3
π
74. y 2x tan0.3x, x 1, x 4, y 0
Approximation In Exercises 65 and 66, determine which value
best approximates the area of the region between the x-axis and
the graph of the function over the given interval. (Make your
selection on the basis of a sketch of the region and not by
performing any calculations.)
(a) 6
π
2
Area In Exercises 71–74, find the area of the region bounded
by the graphs of the equations. Use a graphing utility to verify
your result.
71. y x
1
dt
t
1
3x
−π
−1
1 x
dx
1 x
In Exercises 61– 64, find Fx.
π
2
csc 2 cot 22 d
4
csc x sin x dx
61. F x 2
0.1
1
dx
1 x
x
dx
x1
2
59.
y
0
0.2
54.
sin x
1 cos x
1
1
dx
x ln x
In Exercises 55– 60, use a computer algebra system to find or
evaluate the integral.
55.
70. y y
1
dx
1 x 2
e2
1 ln x2
dx
x
1
2 2
x 2
dx
51.
0 x 1
2
1 cos d
53.
1 sin 49.
69. y tan x
1
5
dx
3x 1
339
The Natural Logarithmic Function: Integration
1
−2
x
1
2
3
4
3 x dx
x
dx
x 2 43
x
dx
x2 4
sec2 x
dx
tan x
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Logarithmic, Exponential, and Other Transcendental Functions
In Exercises 83 – 86, show that the two formulas are equivalent.
83.
p
tan x dx ln cos x C
94. Sales The rate of change in sales S is inversely proportional
to time t t > 1 measured in weeks. Find S as a function of t
if sales after 2 and 4 weeks are 200 units and 300 units,
respectively.
cot x dx ln sin x C
95. Orthogonal Trajectory
cot x dx ln csc x C
85.
(a) Use a graphing utility to graph the equation
2x 2 y 2 8.
sec x dx ln sec x tan x C
(b) Evaluate the integral to find y 2 in terms of x.
y 2 e1x dx
sec x dx ln sec x tan x C
86.
For a particular value of the constant of integration, graph
the result in the same viewing window used in part (a).
csc x dx ln csc x cot x C
(c) Verify that the tangents to the graphs of parts (a) and (b)
are perpendicular at the points of intersection.
csc x dx ln csc x cot x C
96. Graph the function
fkx In Exercises 87–90, find the average value of the function over
the given interval.
8
87. f x 2,
x
4x 1
88. f x ,
x2
2, 4
y
−4 −3 −2 −1
89. f x for k 1, 0.5, and 0.1 on 0, 10. Find lim fkx.
True or False? In Exercises 97–100, determine whether the
statement is true or false. If it is false, explain why or give an
example that shows it is false.
1
97. ln x12 2ln x
Average
value
Average
value
98. ln x dx 1x C
99.
x
x
1 2 3 4
ln x
, 1, e
x
91. Population Growth
rate of
−1
−2
90. f x sec
xk 1
k
k→0
2, 4
y
7
6
5
4
3
2
1
90,000 .
400 3x
Find the average price p on the interval 40 ≤ x ≤ 50.
tan x dx ln sec x C
84.
93. Average Price The demand equation for a product is
1 2 3 4
x
, 0, 2
6
A population of bacteria is changing at a
1
dx ln cx ,
x
2
100.
c0
2
1
dx ln x
1 x
1
ln 2 ln 1 ln 2
101. Graph the function
f x x
1 x2
on the interval 0, .
3000
dP
dt
1 0.25t
(a) Find the area bounded by the graph of f and the line
y 12 x.
where t is the time in days. The initial population (when t 0)
is 1000. Write an equation that gives the population at any time
t, and find the population when t 3 days.
(b) Determine the values of the slope m such that the line
y mx and the graph of f enclose a finite region.
92. Heat Transfer Find the time required for an object to cool
from 300
F to 250
F by evaluating
10
t
ln 2
300
250
1
dT
T 100
where t is time in minutes.
(c) Calculate the area of this region as a function of m.
102. Prove that the function
2x
F x x
1
dt
t
is constant on the interval 0, .