Learning Objectives
- 3.2.1Solve integration problems involving products and powers of and
- 3.2.2Solve integration problems involving products and powers of and
- 3.2.3Use reduction formulas to solve trigonometric integrals.
In this section we look at how to integrate a variety of products of trigonometric functions. These integrals are called trigonometric integrals. They are an important part of the integration technique called trigonometric substitution, which is featured in Trigonometric Substitution. This technique allows us to convert algebraic expressions that we may not be able to integrate into expressions involving trigonometric functions, which we may be able to integrate using the techniques described in this section. In addition, these types of integrals appear frequently when we study polar, cylindrical, and spherical coordinate systems later. Let’s begin our study with products of and
Integrating Products and Powers of sinx and cosx
A key idea behind the strategy used to integrate combinations of products and powers of and involves rewriting these expressions as sums and differences of integrals of the form or After rewriting these integrals, we evaluate them using u-substitution. Before describing the general process in detail, let’s take a look at the following examples.
Example 3.8
Integrating
Evaluate
Solution
Use -substitution and let In this case, Thus,
Checkpoint 3.5
Evaluate
Example 3.9
A Preliminary Example: Integrating Where k is Odd
Evaluate
Solution
To convert this integral to integrals of the form rewrite and make the substitution Thus,
Checkpoint 3.6
Evaluate
In the next example, we see the strategy that must be applied when there are only even powers of and For integrals of this type, the identities
and
are invaluable. These identities are sometimes known as power-reducing identities and they may be derived from the double-angle identity and the Pythagorean identity
Example 3.10
Integrating an Even Power of
Evaluate
Solution
To evaluate this integral, let’s use the trigonometric identity Thus,
Checkpoint 3.7
Evaluate
The general process for integrating products of powers of and is summarized in the following set of guidelines.
Problem-Solving Strategy
Integrating Products and Powers of sin x and cos x
To integrate use the following strategies:
- If is odd, rewrite and use the identity to rewrite in terms of Integrate using the substitution This substitution makes
- If is odd, rewrite and use the identity to rewrite in terms of Integrate using the substitution This substitution makes (Note: If both and are odd, either strategy 1 or strategy 2 may be used.)
- If both and are even, use and After applying these formulas, simplify and reapply strategies 1 through 3 as appropriate.
Example 3.11
Integrating where k is Odd
Evaluate
Solution
Since the power on is odd, use strategy 1. Thus,
Example 3.12
Integrating where k and j are Even
Evaluate
Solution
Since the power on is even and the power on is even we must use strategy 3. Thus,
Since has an even power, substitute
Checkpoint 3.8
Evaluate
Checkpoint 3.9
Evaluate
In some areas of physics, such as quantum mechanics, signal processing, and the computation of Fourier series, it is often necessary to integrate products that include and These integrals are evaluated by applying trigonometric identities, as outlined in the following rule.
Rule: Integrating Products of Sines and Cosines of Different Angles
To integrate products involving and use the substitutions
(3.3)
(3.4)
(3.5)
These formulas may be derived from the sum-of-angle formulas for sine and cosine.
Example 3.13
Evaluating
Evaluate
Solution
Apply the identity Thus,
Integrating Products and Powers of tanx and secx
Before discussing the integration of products and powers of and it is useful to recall the integrals involving and we have already learned:
For most integrals of products and powers of and we rewrite the expression we wish to integrate as the sum or difference of integrals of the form or As we see in the following example, we can evaluate these new integrals by using u-substitution.
Example 3.14
Evaluating
Evaluate
Solution
Start by rewriting as
Media
You can read some interesting information at this website to learn about a common integral involving the secant.
Checkpoint 3.11
Evaluate
We now take a look at the various strategies for integrating products and powers of and
Problem-Solving Strategy
Integrating
To integrate use the following strategies:
- If is even and rewrite and use to rewrite in terms of Let and
- If is odd and rewrite and use to rewrite in terms of Let and (Note: If is even and is odd, then either strategy 1 or strategy 2 may be used.)
- If is odd where and rewrite It may be necessary to repeat this process on the term.
- If is even and is odd, then use to express in terms of Use integration by parts to integrate odd powers of
Example 3.15
Integrating when is Even
Evaluate
Solution
Since the power on is even, rewrite and use to rewrite the first in terms of Thus,
Example 3.16
Integrating when is Odd
Evaluate
Solution
Since the power on is odd, begin by rewriting Thus,
Example 3.17
Integrating where is Odd and
Evaluate
Solution
Begin by rewriting Thus,
For the first integral, use the substitution For the second integral, use the formula.
Example 3.18
Integrating
Integrate
Solution
This integral requires integration by parts. To begin, let and These choices make and Thus,
We now have
Since the integral has reappeared on the right-hand side, we can solve for by adding it to both sides. In doing so, we obtain
Dividing by 2, we arrive at
Checkpoint 3.12
Evaluate
Reduction Formulas
Evaluating for values of where is odd requires integration by parts. In addition, we must also know the value of to evaluate The evaluation of also requires being able to integrate To make the process easier, we can derive and apply the following power reduction formulas. These rules allow us to replace the integral of a power of or with the integral of a lower power of or
Rule: Reduction Formulas for and
(3.6)
(3.7)
The first power reduction rule may be verified by applying integration by parts. The second may be verified by following the strategy outlined for integrating odd powers of
Example 3.19
Revisiting
Apply a reduction formula to evaluate
Solution
By applying the first reduction formula, we obtain
Example 3.20
Using a Reduction Formula
Evaluate
Solution
Applying the reduction formula for we have
Checkpoint 3.13
Apply the reduction formula to
Section 3.2 Exercises
Fill in the blank to make a true statement.
69.
70.
Use an identity to reduce the power of the trigonometric function to a trigonometric function raised to the first power.
71.
72.
Evaluate each of the following integrals by u-substitution.
73.
74.
75.
76.
77.
78.
Compute the following integrals using the guidelines for integrating powers of trigonometric functions. Use a CAS to check the solutions. (Note: Some of the problems may be done using techniques of integration learned previously.)
79.
80.
81.
82.
83.
84.
85.
86.
87.
88.
89.
90.
91.
92.
93.
94.
For the following exercises, find a general formula for the integrals.
95.
96.
Use the double-angle formulas to evaluate the following integrals.
97.
98.
99.
100.
101.
102.
For the following exercises, evaluate the definite integrals. Express answers in exact form whenever possible.
103.
104.
105.
106.
107.
108.
109.
(Round this answer to three decimal places.)
110.
111.
112.
Find the area of the region bounded by the graphs of the equations
113.
Find the area of the region bounded by the graphs of the equations
114.
A particle moves in a straight line with the velocity function Find its position function if
115.
Find the average value of the function over the interval
For the following exercises, solve the differential equations.
116.
The curve passes through point
117.
118.
Find the length of the curve
119.
Find the length of the curve
120.
Find the volume generated by revolving the curve about the x-axis,
For the following exercises, use this information: The inner product of two functions f and g over is defined by Two distinct functions f and g are said to be orthogonal if
121.
Show that are orthogonal over the interval
122.
Evaluate
123.
Integrate
For each pair of integrals, determine which one is more difficult to evaluate. Explain your reasoning.
124.
or
125.
or