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388 CHAPTER 6 Techniques of Integration
6.1 INTEGRATION BY SUBSTITUTION
■ Use the basic integration formulas to find indefinite integrals.
■ Use substitution to find indefinite integrals.
■ Use substitution to evaluate definite integrals.
■ Use integration to solve real-life problems.
Review of Basic Integration Formulas
Each of the basic integration rules you studied in Chapter 5 was derived from a
corresponding differentiation rule. It may surprise you to learn that, although you
now have all the necessary tools for differentiating algebraic, exponential, and
logarithmic functions, your set of tools for integrating these functions is by no
means complete. The primary objective of this chapter is to develop several
techniques that greatly expand the set of integrals to which the basic integration
formulas can be applied.
Basic Integration Formulas
1. Constant Rule: k dx kx C
xn1
n
2. Simple Power Rule n 1: x dx C
n 1
ndu n
3. General Power Rule n 1: u dx u du
dx
un1
n 1 C
4. Simple Exponential Rule: ex dx ex C
u du u
5. General Exponential Rule: e dx e du
dx
eu C
6. Simple Log Rule: 1 dx ln x C
x
dudx 1
7. General Log Rule: dx du
u u
ln u C
As you will see once you work a few integration problems, integration is not
nearly as straightforward as differentiation. A major part of any integration prob-
lem is determining which basic integration formula (or formulas) to use to solve
the problem. This requires remembering the basic formulas, familiarity with
various procedures for rewriting integrands in the basic forms, and lots of practice.
SECTION 6.1 Integration by Substitution 389
STUDY TIP
Integration by Substitution
There are several techniques for rewriting an integral so that it fits one or more of When you use integration by
the basic formulas. One of the most powerful techniques is integration by substitution, you need to realize
substitution. With this technique, you choose part of the integrand to be u and that your integral should contain
then rewrite the entire integral in terms of u. just one variable. For instance,
the integrals
EXAMPLE 1 Integration by Substitution x dx
2
Use the substitution u x 1 to find the indefinite integral. x 1
x and
dx
2 u 1
x 1 du
2
SOLUTION From the substitution u x 1, u
du are in the correct form, but the
x u 1, dx 1, and dxdu. integral
By replacing all instances of x and dx with the appropriate u-variable forms, you x dx
2
obtain u
x u 1 is not.
dx du Substitute for x and dx.
2 2
x 1 u
u 1 du Write as separate fractions.
2 2
u u TRY IT 1
1 1 du Simplify.
2 Use the substitution u x 2
u u
1 to find the indefinite integral.
ln u C Find antiderivative.
u x
dx
2
1 x 2
ln x 1 C. Substitute for u.
x 1
The basic steps for integration by substitution are outlined in the guidelines
below.
Guidelines for Integration by Substitution
1. Let u be a function of x (usually part of the integrand).
2. Solve for x and dx in terms of u and du.
3. Convert the entire integral to u-variable form and try to fit it to one
or more of the basic integration formulas. If none fits, try a different
substitution.
4. After integrating, rewrite the antiderivative as a function of x.
390 CHAPTER 6 Techniques of Integration
DISCOVERY EXAMPLE 2 Integration by Substitution
Suppose you were asked to eval-
uate the integrals below. Which 2
Find x x 1 dx.
one would you choose? Explain
your reasoning. SOLUTION Consider the substitution u x2 1, which produces du 2x dx.
To create 2xdxas part of the integral, multiply and divide by 2.
2
x 1 dx or 12 du
u
2 1 2 12
2
x x 1 dx x 1 2x dx Multiply and divide by 2.
x x 1 dx 2
1 12
u du Substitute for x and dx.
2
32
1 u C Power Rule
2 32
1 32
3u C Simplify.
1 2 32
3x 1 C Substitute for u.
You can check this result by differentiating.
TRY IT 2
2
Find x x 4 dx.
EXAMPLE 3 Integration by Substitution
e3x
Find dx.
1e3x
3x 3x
SOLUTION Consider the substitution u 1 e ,which produces du 3e dx.
3x
To create 3e dxas part of the integral, multiply and divide by 3.
1u du
3x
e 1 1 3x
dx 3e dx Multiply and divide by 3.
3x 3x
1 e 3 1 e
1 1 du Substitute for x and dx.
3 u
1 ln u C Log Rule
3
TRY IT 3 1
3x
e2x 3 ln1 e C Substitute for u.
Find dx.
1e2x
Note that the absolute value is not necessary in the final answer because the quan-
3x
tity 1 e is positive for all values of x.
SECTION 6.1 Integration by Substitution 391
EXAMPLE 4 Integration by Substitution
Find the indefinite integral.
x x 1 dx
SOLUTION Consider the substitution u x 1, which produces du dx and
x u 1.
12
x x 1 dx u 1u du Substitute for x and dx.
32 12
u u du Multiply.
52 32
u u C Power Rule
52 32
2 52 2 32
5x 1 3x1 C Substitute for u.
This form of the antiderivative can be further simplified.
2 52 2 32 6 52 10 32
x 1 x1 C x1 x1 C
5 3 15 15
2 32
15x 1 3x 1 5
C
2 32
15x 1 3x 2 C
You can check this answer by differentiating.
TRY IT 4
Find the indefinite integral.
x x 2 dx
Example 4 demonstrates one of the characteristics of integration by substi-
tution. That is, you can often simplify the form of the antiderivative as it exists
immediately after resubstitution into x-variable form. So, when working the exer-
cises in this section, don’t assume that your answer is incorrect just because it
doesn’t look exactly like the answer given in the back of the text. You may be able
to reconcile the two answers by algebraic simplification.
TECHNOLOGY
If you have access to a symbolic integration utility, try using it to
2
find an antiderivative of fx x x 1 and check your answer
analytically using the substitution u x 1.You can also use the utility
to solve several of the exercises in this section.
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