7
Techniques of Integration
Copyright © Cengage Learning. All rights reserved.
7.2
Trigonometric Integrals
Copyright © Cengage Learning. All rights reserved.
Trigonometric Integrals
In this section we use trigonometric identities to integrate certain combinations of trigonometric functions.
3
Integrals of Powers of Sine and Cosine
4
Integrals of Powers of Sine and Cosine (1 of 4)
We begin by considering integrals in which the integrand is a power of sine, a power of cosine, or a product of these.
5
Example 2
Find
Solution:
We could convert
but we would be left with an expression
in terms of sin x with no extra cos x factor.
Instead, we separate a single sine factor and rewrite the remaining
factor in terms of cos x:
6
Example 2 – Solution
Substituting u = cos x, we have d u = −sin x dx and so
7
Example 3
Evaluate
Solution:
If we write
the integral is no simpler to evaluate. Using the
half-angle formula for
however, we have
8
Example 3 – Solution
Notice that we mentally made the substitution u = 2x when integrating cos 2x.
9
Integrals of Powers of Sine and Cosine (2 of 4)
To summarize, we list guidelines to follow when evaluating integrals of the form
where m ≥ 0 and n ≥ 0 are integers.
10
Integrals of Powers of Sine and Cosine (3 of 4)
Strategy for Evaluating
(a) If the power of cosine is odd (n = 2k + 1), save one cosine factor and use
to express the remaining factors in terms of sine:
Then substitute u = sin x.
(b) If the power of sine is odd (m = 2k + 1), save one sine factor and use
to express the remaining factors in terms of cosine:
11
Integrals of Powers of Sine and Cosine (4 of 4)
Then substitute u = cos x. [Note that if the powers of both sine and cosine are odd, either (a) or (b) can be used.]
(c) If the powers of both sine and cosine are even, use the half-angle identities
It is sometimes helpful to use the identity
12
Integrals of Powers of Secant and Tangent
13
Integrals of Powers of Secant and Tangent (1 of 6)
We can use a similar strategy to evaluate integrals of the form
Since
we can separate a
factor and convert the
remaining (even) power of secant to an expression involving tangent using the
identity
Or, since
we can separate a sec x tan x factor and convert
the remaining (even) power of tangent to secant.
14
Example 5
Evaluate
Solution:
If we separate one
factor, we can express the remaining
factor in terms of tangent using the identity
We can then evaluate the integral by substituting u = tan x so that
15
Example 5 – Solution
16
Integrals of Powers of Secant and Tangent (2 of 6)
The preceding examples demonstrate strategies for evaluating integrals of the form
for two cases, which we summarize here.
Strategy for Evaluating
(a) If the power of secant is even (n = 2k, k ≥ 2), save a factor of
to express the remaining factors in terms of tan x:
Then substitute u = tan x.
17
Integrals of Powers of Secant and Tangent (3 of 6)
(b) If the power of tangent is odd (m = 2k + 1), save a factor of sec x tan x and use
to express the remaining factors in terms of sec x:
Then substitute u = sec x.
18
Integrals of Powers of Secant and Tangent (4 of 6)
For other cases, the guidelines are not as clear-cut. We may need to use identities, integration by parts, and occasionally a little ingenuity.
We will sometimes need to be able to integrate tan x by using the formula given below:
19
Integrals of Powers of Secant and Tangent (5 of 6)
We will also need the indefinite integral of secant:
We could verify Formula 1 by differentiating the right side, or as follows. First we multiply numerator and denominator by sec x + tan x:
20
Integrals of Powers of Secant and Tangent (6 of 6)
If we substitute u = sec x + tan x, then
so the integral becomes
Thus we have
21
Example 7
Find
Solution:
Here only tan x occurs, so we use
to rewrite a
factor in terms of
22
Example 7 – Solution
In the first integral we mentally substituted u = tan x so that
23
Using Product Identities
24
Using Product Identities (1 of 1)
The following product identities are useful in evaluating certain trigonometric integrals.
2 To evaluate the integrals (a)
(b)
(c)
use the corresponding identity:
(a)
(b)
(c)
25
Example 9
Evaluate
Solution:
This integral could be evaluated using integration by parts, but it’s easier to use the identity in Equation 2(a) as follows:
26