# 3.2 Trigonometric Integrals

### Learning Objectives

- Solve integration problems involving products of powers of and
- Integrate products of sines and cosines of different angles.
- Solve integration problems involving products of powers of and
- Use reduction formulas to evaluate 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 *used for integrating functions involving certain root expressions that will be discussed in the next 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 powers of and

### Integrating Products and Powers of sin(*x*) and cos(*x*)

A key idea behind the strategy used to integrate combinations of powers of and involves rewriting these expressions as sums and differences of integrals of the form or that can be evaluated using *u*-substitution. Before describing the general process in detail, let’s take a look at the following examples.

### Evaluating

Evaluate

#### Solution

Make a substitution In this case, Thus,

Evaluate

#### Answer

#### Hint

Take

### A Preliminary Example: Evaluating When is Odd

Evaluate

#### Solution

To convert this integral into a combination of integrals the form rewrite

We now make a substitution , which means that , and obtain

Evaluate

#### Answer

#### Hint

Write and let

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

### Integrating an Even Power of

Evaluate

#### Solution

To evaluate this integral, let’s use the trigonometric identity Thus,

Evaluate

#### Answer

#### Hint

The general process for integrating products of powers of and is summarized in the following set of guidelines.

### Problem-Solving Strategy: Integrating Products of Powers of sin(*x*) and cos(*x*)

To evaluate 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 identities and After applying these formulas, simplify and reapply strategies 2 and 3 to the combination of powers of as appropriate.

### Evaluating When is Odd

Evaluate

#### Solution

Since the power on is odd, use strategy 2.

### Evaluating When and are Even

Evaluate

#### Solution

Since both the powers of and are even we must use strategy 3. Thus,

Since has an even power, we use strategy 3 again and substitute to continue the equalities as follows:

Evaluate

#### Answer

#### Hint

Use strategy 2. Write and substitute

Evaluate

#### Answer

#### Hint

Use strategy 3 and 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 below.

### Integrating Products of Sines and Cosines of Different Angles

To integrate products involving and use the following identities

These formulas may be derived from the sum-of-angle formulas for sine and cosine.

### Evaluating

Evaluate

#### Solution

Apply the identity Thus,

Evaluate

#### Answer

#### Hint

Rewrite

### Integrating Products and Powers of tan(*x*) and sec(*x*)

Before discussing the integration of products of powers of and it is useful to recall the integrals involving and we have already learned:

(Formulas 1 and 2 come directly from the table of indefinite integrals, while formulas 3 and 4 were derived in Section 1.5 Substitution.)

For most integrals of products of 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 appropriate substitution.

### Evaluating

Evaluate

#### Solution

Start by rewriting as If we now let , then , and so

Evaluate

#### Answer

#### Hint

Let and

We now take a look at the various strategies for integrating products of powers of and

### Problem-Solving Strategy: Evaluating

To evaluate 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 express in terms of Let and

(*Note*: If is even and is odd, then either strategy 1 or strategy 2 may be used.) - If is even and is odd, then use to express in terms of Use integration by parts to integrate odd powers of

### Evaluating When is Even

Evaluate

#### Solution

Since the power on is even, rewrite and use to express the first in terms of We now make a substitution , in which case , and we obtain

### Evaluating When is Odd

Evaluate

#### Solution

Since the power of is odd, we begin by rewriting We then notice that , and make a substitution with . With this, we obtain

### Evaluating

Evaluate

#### Solution

Although there is no under the integral, we can still use the strategy outlined above for the case when the power of is odd. For this, we will need to multiply and divide the integrand by :

Hence, using the substitution , we obtain

### Evaluating

Integrate

#### Solution

This integral requires integration by parts. Let and These choices make and Thus,

We now have

We see that the last integral is the same as the original one. Let be a particular antiderivative of . Substituting instead of into the above equality, we get the following equation for .

.

Adding to both sides, we obtain

Dividing by 2, we arrive at

Since an indefinite integral can be found by adding an arbitrary constant to a particular antiderivative of the integrand, we obtain that .

Evaluate

#### Answer

### Reduction Formulas

Evaluating for odd values of 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. They allow us to replace the integral of a power of or with the integral of a lower power of or

### Reduction Formulas for and

The first power reduction rule may be verified by applying integration by parts. The second may be verified by rewriting

### Revisiting

Apply a reduction formula to evaluate

#### Solution

By applying the first reduction formula with , we obtain

### Using a Reduction Formula

Evaluate

#### Solution

Applying the second reduction formula with , we obtain

.

To evaluate , we apply the second reduction formula with , which allows to continue the chain of equalities as follows:

Apply the reduction formula to

#### Answer

### Key Concepts

Integrals of trigonometric functions can be evaluated using various strategies. These strategies include the following.

- Applying trigonometric identities to rewrite the integrand so that it may be evaluated via an apropriate substitution.
- Using integration by parts.
- Applying trigonometric identities to rewrite products of sines and cosines with different arguments as the sum of individual sine and cosine functions.
- Applying reduction formulas.

### Key Equations

**Sine Products**

**Sine and Cosine Products**

**Cosine Products**

**Power Reduction Formula**

**Power Reduction Formula**

### Exercises

Fill in the blank to make a true statement.

**1.**

#### Answer

**2.**

Use an identity to reduce the power of the trigonometric function to a trigonometric function raised to the first power.

**3.**

#### Answer

**4.**

Evaluate each of the following integrals using appropriate substitution.

**5.**

#### Answer

**6.**

**7.**

#### Answer

**8.**

**9.**

#### Answer

**10.**

Compute the following integrals using the guidelines for integrating powers of trigonometric functions. (*Note*: Some of the problems may be done using techniques of integration learned previously.)

**11.**

#### Answer

**12.**

**13.**

#### Answer

**14.**

**15.**

#### Answer

**16.**

**17.**

#### Answer

**18.**

**19.**

(Hint: use integration by parts)

#### Answer

**20.**

**21.**

#### Answer

**22.**

**23.**

#### Answer

**24.**

For the following exercises, evaluate the integrals involving parameter .

**25.**

#### Answer

**26.**

Use the double-angle formulas to evaluate the following integrals.

**27.**

#### Answer

**28.**

**29.**

#### Answer

**30.**

**31.**

#### Answer

**32.**

(Hint: to integrate , rewrite it as and use the formula.)

For the following exercises, evaluate the definite integrals using an appropriate trigonometric formula.

**33.**

#### Answer

**34.**

**35.**

#### Answer

**36.**

**37.**

#### Answer

**38.**

**39.**

#### Answer

**40.**

**41.**

#### Answer

**42.** Find the area of the region bounded by the curves

**43.** Find the area of the region bounded by the curves

#### Answer

1

**44.** A particle moves in a straight line with the velocity function Find its position function if

**45.** Find the average value of the function over the interval

#### Answer

**46.** Find the length of the curve

**47.** Find the length of the curve

#### Answer

**48.** Let be the region below the curve and above the *x*-axis over the interval . Find the volume generated by revolving about the *x*-axis.

**49.** The inner product of two functions *f* and *g* over is defined by Two non-zero functions *f* and *g* are said to be orthogonal if

Show that and are orthogonal over the interval

#### Answer

**50.** Let be real numbers such that . Evaluate

For each pair of integrals, determine which one is easier to evaluate. Explain your reasoning.

**51.** or

#### Answer

The first integral is easier to evaluate as it can be done just using a substitution evaluated , while the second integral is of a complicated type when the power of tangent is even and the power of secant is odd.

**52.** or

### Glossary

- power reduction formula
- a rule that allows an integral of a power of a trigonometric function to be exchanged for an integral involving a lower power

- trigonometric integral
- an integral involving powers and products of trigonometric functions