5. Integration

# 5.7 Integrals Resulting in Inverse Trigonometric Functions

### Learning Objectives

- Integrate functions resulting in inverse trigonometric functions

In this section we focus on integrals that result in inverse trigonometric functions. We have worked with these functions before. Recall from Functions and Graphs that trigonometric functions are not one-to-one unless the domains are restricted. When working with inverses of trigonometric functions, we always need to be careful to take these restrictions into account. Also in Derivatives, we developed formulas for derivatives of inverse trigonometric functions. The formulas developed there give rise directly to integration formulas involving inverse trigonometric functions.

# Integrals that Result in Inverse Sine Functions

Let us begin this last section of the chapter with the three formulas. Along with these formulas, we use substitution to evaluate the integrals. We prove the formula for the inverse sine integral.

### Rule: Integration Formulas Resulting in Inverse Trigonometric Functions

The following integration formulas yield inverse trigonometric functions:

## Proof

Let Then Now let’s use implicit differentiation. We obtain

For Thus, applying the Pythagorean identity we have This gives

Then for we have

□

### Evaluating a Definite Integral Using Inverse Trigonometric Functions

Evaluate the definite integral

We can go directly to the formula for the antiderivative in the rule on integration formulas resulting in inverse trigonometric functions, and then evaluate the definite integral. We have

*** QuickLaTeX cannot compile formula: \begin{array}{}\\ \\ {\int }_{0}^{1}\frac{dx}{\sqrt{1-{x}^{2}}}\hfill & ={ \sin }^{-1}x{|}_{0}^{1}\hfill \\ & ={ \sin }^{-1}1-{ \sin }^{-1}0\hfill \\ & =\frac{\pi }{2}-0\hfill \\ & =\frac{\pi }{2}.\hfill \end{array} *** Error message: Missing # inserted in alignment preamble. leading text: $\begin{array}{} Missing $ inserted. leading text: $\begin{array}{}\\ \\ {\int Extra }, or forgotten $. leading text: $\begin{array}{}\\ \\ {\int } Missing } inserted. leading text: ..._{0}^{1}\frac{dx}{\sqrt{1-{x}^{2}}}\hfill & Extra }, or forgotten $. leading text: ..._{0}^{1}\frac{dx}{\sqrt{1-{x}^{2}}}\hfill & Missing } inserted. leading text: ..._{0}^{1}\frac{dx}{\sqrt{1-{x}^{2}}}\hfill & Extra }, or forgotten $. leading text: ..._{0}^{1}\frac{dx}{\sqrt{1-{x}^{2}}}\hfill & Missing } inserted. leading text: ..._{0}^{1}\frac{dx}{\sqrt{1-{x}^{2}}}\hfill & Extra }, or forgotten $. leading text: ..._{0}^{1}\frac{dx}{\sqrt{1-{x}^{2}}}\hfill & Missing } inserted. leading text: ..._{0}^{1}\frac{dx}{\sqrt{1-{x}^{2}}}\hfill &

Find the antiderivative of

#### Solution

### Finding an Antiderivative Involving an Inverse Trigonometric Function

Evaluate the integral

Substitute Then and we have

Applying the formula with we obtain

Find the indefinite integral using an inverse trigonometric function and substitution for

#### Solution

#### Hint

Use the formula in the rule on integration formulas resulting in inverse trigonometric functions.

### Evaluating a Definite Integral

Evaluate the definite integral

#### Solution

The format of the problem matches the inverse sine formula. Thus,

*** QuickLaTeX cannot compile formula: \begin{array}{}\\ \\ {\int }_{0}^{\sqrt{3}\text{/}2}\frac{du}{\sqrt{1-{u}^{2}}}\hfill & ={ \sin }^{-1}u{|}_{0}^{\sqrt{3}\text{/}2}\hfill \\ & =\left[{ \sin }^{-1}(\frac{\sqrt{3}}{2})\right]-\left[{ \sin }^{-1}(0)\right]\hfill \\ & =\frac{\pi }{3}.\hfill \end{array} *** Error message: Missing # inserted in alignment preamble. leading text: $\begin{array}{} Missing $ inserted. leading text: $\begin{array}{}\\ \\ {\int Extra }, or forgotten $. leading text: $\begin{array}{}\\ \\ {\int } Missing } inserted. leading text: ...ext{/}2}\frac{du}{\sqrt{1-{u}^{2}}}\hfill & Extra }, or forgotten $. leading text: ...ext{/}2}\frac{du}{\sqrt{1-{u}^{2}}}\hfill & Missing } inserted. leading text: ...ext{/}2}\frac{du}{\sqrt{1-{u}^{2}}}\hfill & Extra }, or forgotten $. leading text: ...ext{/}2}\frac{du}{\sqrt{1-{u}^{2}}}\hfill & Missing } inserted. leading text: ...ext{/}2}\frac{du}{\sqrt{1-{u}^{2}}}\hfill & Extra }, or forgotten $. leading text: ...ext{/}2}\frac{du}{\sqrt{1-{u}^{2}}}\hfill & Missing } inserted. leading text: ...ext{/}2}\frac{du}{\sqrt{1-{u}^{2}}}\hfill &

# Integrals Resulting in Other Inverse Trigonometric Functions

There are six inverse trigonometric functions. However, only three integration formulas are noted in the rule on integration formulas resulting in inverse trigonometric functions because the remaining three are negative versions of the ones we use. The only difference is whether the integrand is positive or negative. Rather than memorizing three more formulas, if the integrand is negative, simply factor out −1 and evaluate the integral using one of the formulas already provided. To close this section, we examine one more formula: the integral resulting in the inverse tangent function.

### Finding an Antiderivative Involving the Inverse Tangent Function

Find an antiderivative of

#### Solution

Comparing this problem with the formulas stated in the rule on integration formulas resulting in inverse trigonometric functions, the integrand looks similar to the formula for So we use substitution, letting then and Then, we have

Use substitution to find the antiderivative of

#### Solution

#### Hint

Use the solving strategy from (Figure) and the rule on integration formulas resulting in inverse trigonometric functions.

### Applying the Integration Formulas

Find the antiderivative of

Apply the formula with Then,

### Evaluating a Definite Integral

Evaluate the definite integral

#### Solution

Use the formula for the inverse tangent. We have

*** QuickLaTeX cannot compile formula: \begin{array}{}\\ \\ {\int }_{\sqrt{3}\text{/}3}^{\sqrt{3}}\frac{dx}{1+{x}^{2}}\hfill & ={ \tan }^{-1}x{|}_{\sqrt{3}\text{/}3}^{\sqrt{3}}\hfill \\ & =\left[{ \tan }^{-1}(\sqrt{3})\right]-\left[{ \tan }^{-1}(\frac{\sqrt{3}}{3})\right]\hfill \\ & =\frac{\pi }{6}.\hfill \end{array} *** Error message: Missing # inserted in alignment preamble. leading text: $\begin{array}{} Missing $ inserted. leading text: $\begin{array}{}\\ \\ {\int Extra }, or forgotten $. leading text: $\begin{array}{}\\ \\ {\int } Missing } inserted. leading text: .../}3}^{\sqrt{3}}\frac{dx}{1+{x}^{2}}\hfill & Extra }, or forgotten $. leading text: .../}3}^{\sqrt{3}}\frac{dx}{1+{x}^{2}}\hfill & Missing } inserted. leading text: .../}3}^{\sqrt{3}}\frac{dx}{1+{x}^{2}}\hfill & Extra }, or forgotten $. leading text: .../}3}^{\sqrt{3}}\frac{dx}{1+{x}^{2}}\hfill & Missing } inserted. leading text: .../}3}^{\sqrt{3}}\frac{dx}{1+{x}^{2}}\hfill & Extra }, or forgotten $. leading text: .../}3}^{\sqrt{3}}\frac{dx}{1+{x}^{2}}\hfill & Missing } inserted. leading text: .../}3}^{\sqrt{3}}\frac{dx}{1+{x}^{2}}\hfill &

Evaluate the definite integral

#### Solution

#### Hint

Follow the procedures from (Figure) to solve the problem.

### Key Concepts

- Formulas for derivatives of inverse trigonometric functions developed in Derivatives of Exponential and Logarithmic Functions lead directly to integration formulas involving inverse trigonometric functions.
- Use the formulas listed in the rule on integration formulas resulting in inverse trigonometric functions to match up the correct format and make alterations as necessary to solve the problem.
- Substitution is often required to put the integrand in the correct form.

# Key Equations

**Integrals That Produce Inverse Trigonometric Functions**

In the following exercises, evaluate each integral in terms of an inverse trigonometric function.

**1. **

#### Solution

**2. **

**3. **

#### Solution

**4. **

**5. **

#### Solution

**6. **

In the following exercises, find each indefinite integral, using appropriate substitutions.

**7. **

[hidden-answer a=”877132″]

**8. **

**9. **

#### Solution

**10. **

**11. **

#### Solution

**12. **

**13. **Explain the relationship Is it true, in general, that

#### Solution

So, They differ by a constant.

**14. **Explain the relationship Is it true, in general, that

**15. **Explain what is wrong with the following integral:

#### Solution

is not defined as a real number when

**16. **Explain what is wrong with the following integral:

In the following exercises, solve for the antiderivative of with then use a calculator to graph and the antiderivative over the given interval Identify a value of *C* such that adding *C* to the antiderivative recovers the definite integral

**17. [T]** over

#### Solution

The antiderivative is Taking recovers the definite integral.

**18. [T]** over

**19. [T]** over

#### Solution

The antiderivative is Taking recovers the definite integral.

**20. [T]** over

In the following exercises, compute the antiderivative using appropriate substitutions.

**21.**

#### Solution

**22. **

**23. **

#### Solution

**24. **

**25. **

#### Solution

**26. **

In the following exercises, use a calculator to graph the antiderivative with over the given interval Approximate a value of *C*, if possible, such that adding *C* to the antiderivative gives the same value as the definite integral

**27. [T]** over

#### Solution

The antiderivative is Taking recovers the definite integral over

**28. [T]** over

**29. [T]** over

The general antiderivative is Taking recovers the definite integral.

**30. [T]** over

**31. [T]** over

The general antiderivative is Taking recovers the definite integral.

**32. [T]** over

In the following exercises, compute each integral using appropriate substitutions.

**33. **

#### Solution

**34. **

**35. **

#### Solution

**36. **

**37. **

#### Solution

**38. **

In the following exercises, compute each definite integral.

**39. **

[hidden-answer a=”442625″]

**40. **

**41. **

#### Solution

**42. **

**43. **For compute and evaluate the area under the graph of on

#### Solution

as

**44. **For compute and evaluate the area under the graph of over

**45. **Use the substitution and the identity to evaluate (*Hint:* Multiply the top and bottom of the integrand by )

#### Solution

Using the hint, one has Set Then, and the integral is If one uses the identity then this can also be written

**46. [T]** Approximate the points at which the graphs of and intersect, and approximate the area between their graphs accurate to three decimal places.

**47.** **[T]** Approximate the points at which the graphs of and intersect, and approximate the area between their graphs accurate to three decimal places.

#### Solution

The left endpoint estimate with is 2.796 and these decimals persist for

**48. **Use the following graph to prove that

## Hint

Substitute