# 3.3 Trigonometric Substitution

### Learning Objectives

- Solve integration problems involving the square root of a sum or difference of two squares.

In this section, we explore integrals containing expressions of the form and where the values of are positive. We have already encountered and evaluated integrals containing some expressions of this type, but many still remain inaccessible. The technique of trigonometric substitution comes in very handy when evaluating these integrals. This technique uses substitution to rewrite these integrals as trigonometric integrals.

### Integrals Involving

Before developing a general strategy for integrals containing consider the integral This integral cannot be evaluated using any of the techniques we have discussed so far. However, if we make the substitution where , we have After substituting into the integral, we get

After simplifying, we have

Because we obtain

Since , we have that and so which implies

At this point, we can evaluate the integral using the techniques developed for integrating powers of products of trigonometric functions. Before completing this example, let’s take a look at the general theory behind this idea.

To evaluate integrals involving we make the substitution and then Let’s see why this actually makes sense. The domain of is Thus, Consequently, Since the range of over is there is a unique angle satisfying so that or equivalently, so that If we substitute into we get

Since on and From this discussion, we can see that by making the substitution we are able to convert an integral involving a radical into an integral involving trigonometric functions. After we evaluate the integral, we can convert the solution back to an expression involving To see how to do this, let’s begin by assuming that In this case, Since we can draw the reference triangle to assist in expressing the values of and the remaining trigonometric functions in terms of It can be shown that this triangle actually produces the correct values of the trigonometric functions evaluated at for all satisfying It is useful to observe that the expression actually appears as the length of one side of the triangle. Last, should appear by itself, we use

The essential part of this discussion is summarized in the following problem-solving strategy.

### Problem-Solving Strategy: Integrating Expressions Involving

- It is a good idea to make sure the integral cannot be evaluated easily in another way. For example, although this method can be applied to integrals of the form and they can each be integrated directly by a simple substitution.
- Make the substitution and

*Note*: This substitution yields - Simplify the expression.
- Evaluate the integral using techniques from the section on trigonometric integrals.
- Use the reference triangle from Figure 1 to rewrite the result in terms of You may also need to use some trigonometric identities and the relationship

The following example demonstrates the application of this problem-solving strategy.

### Integrating an Expression Involving

Evaluate

#### Solution

Begin by making the substitutions and Since we can construct the reference triangle shown in the following figure.

Thus,

### Integrating an Expression Involving

Evaluate

#### Solution

First make the substitutions and Since we can construct the reference triangle shown in Figure 3 below.

Thus,

In the next example, we see that we sometimes have a choice of methods.

### Integrating an Expression Involving Two Ways

Evaluate two ways: first by using the substitution and then by using a trigonometric substitution.

#### Solution

**Method 1**

Let and hence Thus, In this case, the integral becomes

**Method 2**

Let In this case, Using this substitution, we have

Rewrite the integral using the appropriate trigonometric substitution (do not evaluate the integral).

#### Answer

#### Hint

Substitute and

### Integrating Expressions Involving

For integrals containing let’s first consider the domain of this expression. Since is defined for all real values of we restrict our choice to those trigonometric functions that have a range of all real numbers. Thus, our choice is restricted to selecting either or Either of these substitutions would actually work, but the standard substitution is or, equivalently, With this substitution, we make the assumption that so that we also have The procedure for using this substitution is outlined in the following problem-solving strategy.

### Problem-Solving Strategy: Integrating Expressions Involving

- Check to see whether the integral can be evaluated easily by using another method. In some cases, it is more convenient to use an alternative method.
- Substitute and This substitution yields

(Since and over this interval, ) - Simplify the expression.
- Evaluate the integral using techniques from the section on trigonometric integrals.
- Use the reference triangle from to rewrite the result in terms of You may also need to use some trigonometric identities and the relationship

(*Note*: The reference triangle is based on the assumption that however, the trigonometric ratios produced from the reference triangle are the same as the ratios for which

### Integrating an Expression Involving

Evaluate and check the solution by differentiating.

#### Solution

Begin with the substitution and Since draw the reference triangle in the following figure.

Thus,

To check the solution, differentiate:

Since for all values of we could rewrite if desired.

### Evaluating Using a Different Substitution

Use the substitution to evaluate

#### Solution

Because has a range of all real numbers, and we may also use the substitution to evaluate this integral. In this case, Consequently,

## Analysis

This answer looks quite different from the answer obtained using the substitution To see that the solutions are the same, set Then that is,

After multiplying both sides by and rewriting, this equation becomes:

Use the quadratic equation formula to solve for

Simplifying, we have:

Since it must be the case that Therefore,

At last, we obtain

After we make the final observation that, since

we see that the two different methods produced the same solutions.

### Finding an Arc Length

Find the length of the curve over the interval

#### Solution

Because the arc length is given by

To evaluate this integral, use the substitution and We also need to change the limits of integration. If then and if then Thus,

Rewrite by using a substitution involving

#### Answer

#### Hint

Use and

### Integrating Expressions Involving

The domain of the expression is Thus, either or Hence, or Since these intervals correspond to the range of on the set it makes sense to use the substitution or, equivalently, where or The corresponding substitution for is The procedure for using this substitution is outlined in the following problem-solving strategy.

### Problem-Solving Strategy: Integrals Involving

- Check to see whether the integral cannot be evaluated using another method. If so, we may wish to consider applying an alternative technique.
- Substitute and This substitution yields

For we have , which implies that , and so while for , implying that , and hence - Simplify the expression.
- Evaluate the integral using techniques from the section on trigonometric integrals.
- Use the reference triangles to rewrite the result in terms of You may also need to use some trigonometric identities and the relationship

(*Note*: We need both reference triangles, since the values of some of the trigonometric ratios are different depending on whether or )

### Finding the Area of a Region

Find the area of the region between the graph of and the *x*-axis over the interval

#### Solution

First, sketch a rough graph of the region described in the problem, as shown in the following figure.

We can see that the area is To evaluate this definite integral, substitute and We must also change the limits of integration. If then and hence If then After making these substitutions and simplifying, we have

Evaluate Assume that

#### Answer

#### Hint

Substitute and

### Key Concepts

- For integrals involving use the substitution and
- For integrals involving use the substitution and
- For integrals involving substitute and

### Exercises

Simplify the following expressions by writing each one using a single trigonometric function.

**1.**

#### Answer

**2.**

**3.**

#### Answer

**4.**

Use the technique of completing the square to express each quadratic polynomial in the form .

**5.**

#### Answer

**6.**

**7.**

#### Answer

Evaluate the following integrals using the method of trigonometric substitution.

**8.**

**9.** ()

#### Answer

**10.**

**11.**

#### Answer

**12.**

**13.**

#### Answer

**14.**

**15.**

#### Answer

(Hint: .)

**16.**

**17.**

#### Answer

**18.**

(Hint: factor a power of out of the root.)

**19.**

(Hint: when factoring a power of out of the root, be careful with the signs.)

#### Answer

**20.**

**21.**

#### Answer

**22.**

**23.**

#### Answer

**24.**

**25.**

#### Answer

**26.**

**27.**

#### Answer

In the following exercises, use the substitutions or Express the final answers in terms of the variable *x.*

**28.**

**29.**

#### Answer

**30.**

**31.**

#### Answer

**32.**

**33.**

#### Answer

Combine the technique of completing the square with a trignometric substitution to evaluate the following integrals.

**34.**

**35.**

#### Answer

**36.**

**37.**

#### Answer

**38.**

**39.** Evaluate the integral using geometry.

#### Answer

area of a semicircle with radius 3

**40.** Find the area enclosed by the ellipse

**41.** Evaluate the integral using two different substitutions. First, let and evaluate using trigonometric substitution. Second, let and use trigonometric substitution. Are the answers the same?

#### Answer

and ; these answers are the same since and is a constant.

**42.** Evaluate the integral using the substitution Next, evaluate the same integral using the substitution Show that the results are equivalent.

**43.** Evaluate the integral using the form Next, evaluate the same integral using Are the results the same?

#### Answer

is the result using either method.

**44.** State the method of integration you would use to evaluate the integral Why did you choose this method?

**45.** State the method of integration you would use to evaluate the integral Why did you choose this method?

#### Answer

Use trigonometric substitution

**46.** Find the area bounded by

**47.** During each cycle, the velocity *v* (in feet per second) of a robotic welding device is given by where *t* is time in seconds. Find the expression for the displacement *s* (in feet) as a function of *t* if when

#### Answer

**48.** An oil storage tank can be described as the volume generated by revolving the area bounded by about the *x*-axis. Find the volume of the tank (in cubic meters).

**49.** The region bounded by the graph of and the *x*-axis between and is revolved about the *x-*axis. Find the volume of the solid that is generated.

#### Answer

**50.** Find the length of the arc of the curve over the interval

**51.** Find the length of the curve between and

#### Answer

**52.** Find the area of the surface generated by revolving the curve from about the *y*-axis.

### Glossary

- trigonometric substitution
- an integration technique that converts an algebraic integral containing expressions of the form or into a trigonometric integral