5.7 Integrals 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.

Proof

Let $y = sin^{- 1} \frac{x}{a} .$ Then $a sin y = x .$ Now let’s use implicit differentiation. We obtain

\frac{d}{d x} (a sin y) a cos y \frac{d y}{d x} \frac{d y}{d x} = = = \frac{d}{d x} (x) 1 \frac{1}{a cos y} .

For $- \frac{π}{2} \leq y \leq \frac{π}{2}, cos y \geq 0.$ Thus, applying the Pythagorean identity we have This gives

\frac{1}{a cos y} = \frac{1}{a 1 - sin ^{2} y} = \frac{1}{a ^{2} - a ^{2} sin ^{2} y} = \frac{1}{a ^{2} - x ^{2}} .

Then for $- a \leq x \leq a,$ and generalizing to u, we have

\int \frac{1}{a ^{2} - u ^{2}} d u = sin^{- 1} (\frac{u}{a}) + C .

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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.

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

\int \frac{d u}{a ^{2} - u ^{2}} = sin^{- 1} (\frac{u}{a}) + C