416
CHAPTER 6
6.4
Techniques of Integration
I N T E G R AT I O N TA B L E S A N D C O M P L E T I N G T H E S Q U A R E
■
■
■
Use integration tables to find indefinite integrals.
Use reduction formulas to find indefinite integrals.
Use completing the square to find indefinite integrals.
Integration Tables
So far in this chapter, you have studied three integration techniques to be used
along with the basic integration formulas. Certainly these techniques and formulas do not cover every possible method for finding an antiderivative, but they do
cover most of the important ones.
In this section, you will expand the list of integration formulas to form a table
of integrals. As you add new integration formulas to the basic list, two effects
occur. On one hand, it becomes increasingly difficult to memorize, or even
become familiar with, the entire list of formulas. On the other hand, with a longer
list you need fewer techniques for fitting an integral to one of the formulas on the
list. The procedure of integrating by means of a long list of formulas is called
integration by tables. (The table in this section constitutes only a partial listing
of integration formulas. Much longer lists exist, some of which contain several
hundred formulas.)
Integration by tables should not be considered a trivial task. It requires
considerable thought and insight, and it often requires substitution. Many people
find a table of integrals to be a valuable supplement to the integration techniques
discussed in the first three sections of this chapter. We encourage you to gain
competence in the use of integration tables, as well as to continue to improve in
the use of the various integration techniques. In doing so, you should find that a
combination of techniques and tables is the most versatile approach to integration.
Each integration formula in the table on the next three pages can be
developed using one or more of the techniques you have studied. You should try
to verify several of the formulas. For instance, Formula 4
C
u
1
a
du 2
ln a bu
2
a bu
b a bu
Formula 4
can be verified using partial fractions, Formula 17
a bu
u
du 2a bu a
1
du
ua bu
Formula 17
can be verified using integration by parts, and Formula 37
1
du u ln1 e u C
1 eu
Formula 37
can be verified using substitution.
STUDY
TIP
A symbolic integration utility consists, in part, of a database of integration
tables. The primary difference between using a symbolic integration utility and
using a table of integrals is that with a symbolic integration utility the computer
searches through the database to find a fit. With a table of integrals, you must
do the searching.
SECTION 6.4
Integration Tables and Completing the Square
417
In the table of integrals below and on the next two pages, the formulas have
been grouped into eight different types according to the form of the integrand.
Forms involving u n
Forms involving a bu
Forms involving a bu
Forms involving u2 ± a2
Forms involving u2 a2
Forms involving a2 u2
Forms involving e u
Forms involving ln u
Table of Integrals
Forms involving u n
1.
2.
u n du u n1
C,
n1
n 1
1
du ln u C
u
Forms involving a bu
3.
4.
5.
6.
7.
8.
9.
10.
11.
12.
13.
u
1
du 2 bu a ln a bu C
a bu
b
C
u
1
a
du 2
ln a bu
2
a bu
b a bu
u
1
1
a
du 2
C, n 1, 2
a bu n
b n 2a bun2 n 1a bun1
C
u2
1
bu
du 3 2a bu a2 ln a bu
a bu
b
2
C
u2
1
a2
du 3 bu 2a ln a bu
2
a bu
b
a bu
C
u2
1
2a
a2
du 3
ln a bu
3
a bu
b a bu 2a bu2
u2
1
1
2a
a2
du 3
C, n 1, 2, 3
n
n3
n2
a bu
b n 3a bu
n 2a bu
n 1a bun1
1
1
u
du ln
C
ua bu
a a bu
1
1
1
1
u
du ln
ua bu2
a a bu a a bu
1
1 1 b
u
du ln
a bu
a u a a bu
u2
C
C
1
1 a 2bu
2b
u
du 2
ln
u2a bu2
a ua bu
a
a bu
C
418
Techniques of Integration
CHAPTER 6
Table of Integrals (continued )
Forms involving a bu
14.
15.
16.
17.
18.
19.
20.
u na bu du 22.
23.
24.
25.
26.
27.
28.
a bu a
1
1
du ln
C,
a
a bu a
ua bu
un1a bu du
a > 0
a bu
1
1
2n 3b
du n1
an 1 u
2
bu
una
a bu
1
du
ua bu
du 2a bu a
u
a bu
du un
u
1
a bu32 2n 5b
an 1
un1
2
du a bu
22a bu
a bu C
3b 2
un
2
du una bu na
2n 1b
a bu
Forms involving u2 ± a2,
21.
2
u n a bu32 na
b2n 3
u2 ± a2 du u
1
u n1a
a bu
un1
u n1
du
a bu
1
u2u2 ± a2u2 ± a2 a4 ln u u2 ± a2 C
8
du u2 a2 a ln
a u2 a2
C
u
1
du ln u u2 ± a2 C
u2 ± a2
1 a u2 a2
du
ln
C
a
u
uu2 a2
1
u2
1
du uu2 ± a2 a2 ln u u2 ± a2 C
2
2
2
u ± a
1
u2u2
±
a2
du u2 ± a2
a2 u
C
n1
n1
a > 0
u2
du ,
du ,
u2 ± a2
du ln u u2 ± a2 C
u
u ± a2
2
bu
1
uu2 ± a2 ± a2 ln u u2 ± a2 C
2
u2u2 ± a2 du u2 a2
SECTION 6.4
Integration Tables and Completing the Square
Table of Integrals (continued )
Forms involving u2 a2,
29.
30.
1
du u2 a2
a > 0
32.
33.
1
1
u
1
du 2
2n 3
du , n 1
u2 a2 n
2a n 1 u2 a2n1
u2 a2n1
Forms involving a2 u2 ,
31.
1
1
ua
du ln
C
a2 u2
2a u a
u
a2
u2
a > 0
du a2 u2 a ln
1 a a2 u2
du ln
C
2
2
a
u
ua u
1
1
du u2a2 u2
a2
a2u
u2
a a2 u2
C
u
C
Forms involving e u
34.
35.
36.
37.
38.
e u du e u C
ue u du u 1e u C
u ne u du u ne u n
u n1e u du
1
du u ln1 eu C
1 eu
1
1
du u ln1 enu C
1 enu
n
Forms involving ln u
39.
40.
41.
42.
43.
ln u du u1 ln u C
u ln u du u2
1 2 ln u C
4
un ln u du un1
1 n 1 ln u C,
n 12
ln u2 du u
2 2 ln u ln u2 C
ln un du uln un n ln un1 du
n 1
419
420
CHAPTER 6
Techniques of Integration
T E C H N O L O G Y
Throughout this section,
remember that a symbolic
integration utility can be used
instead of integration tables. If you
have access to such a utility, try
using it to find the indefinite integrals in Examples 1 and 2.
EXAMPLE 1
Find
x
x 1
Using Integration Tables
dx.
Because the expression inside the radical is linear, you should consider forms involving a bu, as in Formula 19.
SOLUTION
u
22a bu
a bu C
du 3b 2
a bu
Formula 19
Using this formula, let a 1, b 1, and u x. Then du dx, and you obtain
x
22 x
x 1 C
dx 3
x 1
2
2 xx 1 C.
3
TRY
IT
1
Use the integration table to find
EXAMPLE 2
Find
x
2 x
Substitute values of
a, b, and u.
Simplify.
dx.
Using Integration Tables
xx 4 9 dx.
Because it is not clear which formula to use, you can begin by
letting u x2 and du 2x dx. With these substitutions, you can write the integral as shown.
SOLUTION
1
2
1
2
xx 4 9 dx x22 92x dx
Multiply and divide by 2.
u2 9 du
Substitute u and du.
Now, it appears that you can use Formula 21.
u2 a2 du Letting a 3, you obtain
TRY
IT
2
Use the integration table to find
x2 16
x
xx4 9 dx 1
uu2 a2 a2 ln u u2 a2 C
2
1
2
u2 a2 du
C
1 1
uu2 a2 a2 ln u u2 a2
2 2
1 2 4
x x 9 9 ln x 2 x 4 9 C.
4
dx.
SECTION 6.4
EXAMPLE 3
Find
421
Integration Tables and Completing the Square
Using Integration Tables
1
dx.
xx 1
Considering forms involving a bu, where a 1, b 1, and
u x, you can use Formula 15.
SOLUTION
So,
a > 0
TRY
a bu a
1
1
du ln
C
a
a bu a
ua bu
x 1 1
C.
ln
x 1 1
1
dx xx 1
EXAMPLE 4
2
Evaluate
0
SOLUTION
a bu a
1
1
du ln
C,
ua bu
a
a bu a
IT
3
Use the integration table to find
1
dx.
x2 4
Using Integration Tables
x
dx.
1 ex 2
Of the forms involving e u, Formula 37
1
du u ln1 e u C
1 eu
seems most appropriate. To use this formula, let u x2 and du 2x dx.
x
1
dx 1 ex2
2
1
1
1
2x dx du
1 ex 2
2 1 eu
1
u ln1 eu C
2
1
2
x2 ln1 ex C
2
1
2
x2 ln1 ex C
2
y
2
y=
x
2
1 + e −x
1
So, the value of the definite integral is
2
0
x
1
2
dx x2 ln1 ex 1 ex2
2
2
0
x
1.66
as shown in Figure 6.10.
TRY
IT
4
FIGURE 6.10
1
Use the integration table to evaluate
1
0
x2
dx.
1 e x3
2
422
CHAPTER 6
Techniques of Integration
Reduction Formulas
Several of the formulas in the integration table have the form
f x dx gx hx dx
where the right side contains an integral. Such integration formulas are called
reduction formulas because they reduce the original integral to the sum of a
function and a simpler integral.
ALGEBRA
REVIEW
For help with the algebra in
Example 5, see Example 2(b) in
the Chapter 6 Algebra Review
on page 447.
EXAMPLE 5
Find
x2e x dx.
SOLUTION
Using a Reduction Formula
Using Formula 36
u neu du u neu n
un1eu du
you can let u x and n 2. Then du dx, and you can write
x2e x dx x2e x 2
xe x dx.
Then, using Formula 35
ueu du u 1eu C
you can write
x2e x dx x2e x 2
TRY
xe x dx
x2e x 2x 1e x C
x2e x 2xe x 2e x C
e xx2 2x 2 C.
IT
5
Use the integration table to find the indefinite integral
ln x2 dx.
T E C H N O L O G Y
You have now studied two ways to find the indefinite integral in
Example 5. Example 5 uses an integration table, and Example 4
in Section 6.2 uses integration by parts. A third way would be to use a
symbolic integration utility.
SECTION 6.4
Integration Tables and Completing the Square
Completing the Square
Many integration formulas involve the sum or difference of two squares. You can
extend the use of these formulas by an algebraic procedure called completing the
square. This procedure is demonstrated in Example 6.
EXAMPLE 6
Completing the Square
Find the indefinite integral.
1
dx
x2 4x 1
SOLUTION
x2
Begin by writing the denominator as the difference of two squares.
4x 1 x2 4x 4 4 1
x 22 3
So, you can rewrite the original integral as
1
dx x2 4x 1
1
dx.
x 22 3
Considering u x 2 and a 3, you can apply Formula 29
1
1
ua
du ln
C
u2 a2
2a u a
to conclude that
1
dx x 4x 1
2
TA K E
1
dx
x 22 3
1
du
2
u a2
1
ua
ln
C
2a u a
x 2 3
1
ln
C.
23 x 2 3
A N O T H E R
L O O K
Using Integration Tables
Which integration formulas on pages 417–419 would you use to find each
indefinite integral? Explain your choice of u for each integral.
a.
ex
dx
ex 1
b.
1
dx
ex 1
c.
1
dx
e 2x 1
Use a symbolic integration utility to check that your choices of u were correct.
TRY
IT
Find
1
dx.
x2 6x 1
6
423
424
Techniques of Integration
CHAPTER 6
P R E R E Q U I S I T E
R E V I E W 6 . 4
The following warm-up exercises involve skills that were covered in earlier sections. You will
use these skills in the exercise set for this section.
In Exercises 1–4, expand the expression.
1. x 42
2. x 12
1 2
2
3. x 2 1
4. x 3 In Exercises 5–8, write the partial fraction decomposition for the expression.
5.
4
xx 2
6.
3
xx 4
7.
x4
x 2 x 2
8.
3x2 4x 8
xx 2x 1
In Exercises 9 and 10, use integration by parts to find the indefinite integral.
9.
10.
2xe x dx
3x2 ln x dx
E X E R C I S E S
6 . 4
In Exercises 1–8, use the indicated formula from the table of integrals in this section to find the indefinite integral.
1.
2.
3.
4.
5.
6.
7.
8.
x
dx, Formula 4
2 3x2
In Exercises 9–34, use the table of integrals in this section to find
the indefinite integral.
9.
1
dx, Formula 11
x2 3x2
11.
x
dx, Formula 19
2 3x
13.
4
dx, Formula 29
x2 9
15.
2x
x 4 9
dx, Formula 25
x2x2 9 dx, Formula 22
2
17.
19.
x3e x dx, Formula 35
21.
x
dx, Formula 37
1 ex2
23.
1
dx
x1 x
1
10.
dx
12.
1
dx
x4 x2
14.
x ln x dx
16.
6x
2 dx
1 e3x
18.
xx 4 4 dx
20.
t2
dt
2 3t3
22.
xx2 9
s
s23
s
ds
24.
1
dx
x1 x2
1
x2 1
x2 9
x2
dx
dx
x2ln x32 dx
1
dx
1 ex
x
dx
x4 9
3 4t
t
dt
3 x2 dx
SECTION 6.4
25.
27.
29.
31.
33.
x2
dx
3 2x5
26.
1
dx
x21 x2
28.
x2 ln x dx
30.
x2
dx
3x 52
32.
ln x
dx
x4 3 ln x
34.
1
dx
x2x2 4
2x
dx
1 3x2
x
49. (a) x2 6x
2
36. y 2
, y 0, x 0, x 1
1 e 4x
55.
37. y x
, y 0, x 2
1 ex2
57.
38. y e x
, y 0, x 1, x 2
1 e 2x
59.
1
, y 0, x 1, x 4
40. y x1 2x (b) x 2 16x 1
(c) x 4 2x2 5
(c) x 4 8x 2 1
(d) 3 2x x2
(d) 9x 2 36x 1
(b) 3x2 12x 9
(b) x 2 4x 1
(c) x2 2x
(c) 1 8x x 2
(d) 9 8x x2
(d) 6x x 2
1
dx
x2 6x 8
41.
0
5
42.
0
4
43.
0
4
44.
x
dx
5 2x
x
dx
4 x2
x
dx
x 4 2x2 2
46.
47.
48.
x 2e x dx
x2 6x dx
7 6x x2
x3
dx
xx 4 4x2 5
dx
x2 2
utility to graph the growth function. Use the table of integrals to
find the average value of the growth function over the interval,
where N is the size of a population and t is the time in days.
61. N 50
,
1 e 4.81.9t
62. N 375
,
1 e 4.200.25t
In Exercises 45–48, find the indefinite integral (a) using the integration table and (b) using the specified method.
1
dx
x2 4x 5
Population Growth In Exercises 61 and 62, use a graphing
R 10,000 1 x ln x dx
1
45.
60.
3, 4
21, 28
63. Revenue The revenue (in dollars per year) for a new
product is modeled by
6
dx
1 e0.5x
Integral
54.
1
dx 56.
x 1x2 2x 2
1
dx
58.
2x2 4x 6
In Exercises 41–44, evaluate the definite integral.
5
52. (a) 16x 2 96x 3
In Exercises 53–60, complete the square and then use the
integration table to find the indefinite integral.
53.
39. y x2x2 4 , y 0, x 5
(b) x2 8x 9
51. (a) 4x 12x 15
, y 0, x 8
x 1
50. (a) x 2 4x
2
ln x 3 dx
In Exercises 35–40, use the integration table to find the exact area
of the region bounded by the graphs of the equations. Then use
a graphing utility to graph the region and approximate the area.
35. y In Exercises 49–52, complete the square to express each
polynomial as the sum or difference of squares.
xe x dx
1
dx
2x22x 12
425
Integration Tables and Completing the Square
Method
64. Consumer and Producer Surpluses Find the consumer surplus and the producer surplus for a product with
the given demand and supply functions.
Demand: p Integration by parts
1
dx
x2x 1
Partial fractions
1
dx
x2 75
Partial fractions
where t is the time in years. Estimate the total revenue from
sales of the product over its first 2 years on the market.
Integration by parts
x 4 ln x dx
1
1 0.1t 212
60
x2 81
,
Supply: p x
3
65. Profit The net profits P (in billions of dollars per year) for
Hershey Foods from 2000 through 2003 can be modeled by
P 0.04t 0.3,
10 ≤ t ≤ 13
where t is the time in years, with t 10 corresponding to
2000. Find the average net profit over that time period.
(Source: Hershey Foods Corp.)