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Mathematics LibreTexts

5.7: Integrals Resulting in Inverse Trigonometric Functions

  • Gilbert Strang & Edwin “Jed” Herman
  • OpenStax

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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, 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, we previously 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 Trigonometric 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:

dua2u2=sin1(ua)+Cdua2+u2=1atan1(ua)+Cduuu2a2=1asec1(|u|a)+C

Proof of the first formula

Let y=sin1xa. Then asiny=x. Now using implicit differentiation, we obtain

ddx(asiny)=ddx(x)

acosydydx=1

dydx=1acosy.

For π2yπ2,cosy0. Thus, applying the Pythagorean identity sin2y+cos2y=1, we have cosy=1sin2y. This gives

1acosy=1a1sin2y=1a2a2sin2y=1a2x2.

Then for axa, we have

1a2u2du=sin1(ua)+C.

Example 5.7.1: Evaluating a Definite Integral Using Inverse Trigonometric Functions

Evaluate the definite integral

1/20dx1x2.

Solution

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

1/20dx1x2=sin1x|1/20=sin112sin10=π60=π6.

Note that since the integrand is simply the derivative of sin1x, we are really just using this fact to find the antiderivative here.

Exercise 5.7.1

Find the indefinite integral using an inverse trigonometric function and substitution for dx9x2.

Hint

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

Answer

dx9x2=sin1(x3)+C

In many integrals that result in inverse trigonometric functions in the antiderivative, we may need to use substitution to see how to use the integration formulas provided above.

Example 5.7.2: Finding an Antiderivative Involving an Inverse Trigonometric Function using substitution

Evaluate the integral

dx49x2.

Solution

Substitute u=3x. Then du=3dx and we have

dx49x2=13du4u2.

Applying the formula with a=2, we obtain

dx49x2=13du4u2=13sin1(u2)+C=13sin1(3x2)+C.

Exercise 5.7.2

Find the antiderivative of dx116x2.

Hint

Substitute u=4x.

Answer

dx116x2=14sin1(4x)+C

Example 5.7.3: Evaluating a Definite Integral

Evaluate the definite integral

3/20du1u2.

Solution

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

3/20du1u2=sin1u|3/20=[sin1(32)][sin1(0)]=π3.

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.

Example 5.7.4: Finding an Antiderivative Involving the Inverse Tangent Function

Find the antiderivative of 19+x2dx.

Solution

Apply the formula with a=3. Then,

dx9+x2=13tan1(x3)+C.

Exercise 5.7.3

Find the antiderivative of dx16+x2.

Hint

Follow the steps in Example 5.7.4.

Answer

dx16+x2=14tan1(x4)+C

Example 5.7.5: Applying the Integration Formulas WITH SUBSTITUTION

Find an antiderivative of 11+4x2dx.

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 tan1u+C. So we use substitution, letting u=2x, then du=2dx and 12du=dx.Then, we have

1211+u2du=12tan1u+C=12tan1(2x)+C.

Exercise 5.7.4

Use substitution to find the antiderivative of dx25+4x2.

Hint

Use the solving strategy from Example 5.7.5 and the rule on integration formulas resulting in inverse trigonometric functions.

Answer

dx25+4x2=110tan1(2x5)+C

Example 5.7.6: Evaluating a Definite Integral

Evaluate the definite integral 33/3dx1+x2.

Solution

Use the formula for the inverse tangent. We have

33/3dx1+x2=tan1x|33/3=[tan1(3)][tan1(33)]=π3π6=π6.

Exercise 5.7.5

Evaluate the definite integral 20dx4+x2.

Hint

Follow the procedures from Example 5.7.6 to solve the problem.

Answer

20dx4+x2=π8

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

dua2u2=sin1(ua)+C

dua2+u2=1atan1(ua)+C

duuu2a2=1asec1(|u|a)+C

  • Gilbert Strang (MIT) and Edwin “Jed” Herman (Harvey Mudd) with many contributing authors. This content by OpenStax is licensed with a CC-BY-SA-NC 4.0 license. Download for free at http://cnx.org.

  • Includes some added textual clarifications and edits by Paul Seeburger (Monroe Community College)

This page titled 5.7: Integrals Resulting in Inverse Trigonometric Functions is shared under a CC BY-NC-SA 4.0 license and was authored, remixed, and/or curated by Gilbert Strang & Edwin “Jed” Herman (OpenStax) via source content that was edited to the style and standards of the LibreTexts platform.

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