3.2.1. Solve integration problems involving products and powers of and
3.2.2. Solve integration problems involving products and powers of and
3.2.3. Use 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.
Use -substitution and let In this case, Thus,
A Preliminary Example: Integrating Where k is Odd
To convert this integral to integrals of the form rewrite and make the substitution Thus,
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
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
Integrating an Even Power of
To evaluate this integral, let’s use the trigonometric identity Thus,
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.
Integrating where k is Odd
Since the power on is odd, use strategy 1. Thus,
Integrating where k and j are Even
Since the power on is even and the power on is even we must use strategy 3. Thus,
Since has an even power, substitute
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
These formulas may be derived from the sum-of-angle formulas for sine and cosine.
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.
Start by rewriting as
You can read some interesting information at this website to learn about a common integral involving the secant.
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.
Integrating when is Even
Since the power on is even, rewrite and use to rewrite the first in terms of Thus,
Integrating when is Odd
Since the power on is odd, begin by rewriting Thus,
Integrating where is Odd and
Begin by rewriting Thus,
For the first integral, use the substitution For the second integral, use the formula.
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
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
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
Apply a reduction formula to evaluate
By applying the first reduction formula, we obtain
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.)