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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.

Example 5.52

Finding an Antiderivative Involving the Inverse Tangent Function

Evaluate the integral \(\int \frac{1}{1 + 4 x^{2}} d x .\)

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 \(\text{tan}^{−1} u + C .\) So we use substitution, letting \(u = 2 x ,\) then \(d u = 2 d x\) and \(1 / 2 d u = d x .\) Then, we have

\[\frac{1}{2} \int \frac{1}{1 + u^{2}} d u = \frac{1}{2} \text{tan}^{−1} u + C = \frac{1}{2} \text{tan}^{−1} \left(\right. 2 x \left.\right) + C .\]

Checkpoint 5.42

Use substitution to find the antiderivative \(\int \frac{d x}{25 + 4 x^{2}} .\)

Example 5.53

Applying the Integration Formulas

Evaluate the integral \(\int \frac{1}{9 + x^{2}} d x .\)

Solution

Apply the formula with \(a = 3 .\) Then,

\[\int \frac{d x}{9 + x^{2}} = \frac{1}{3} \text{tan}^{−1} \left(\right. \frac{x}{3} \left.\right) + C .\]

Checkpoint 5.43

Evaluate the integral \(\int \frac{d x}{16 + x^{2}} .\)

Example 5.54

Evaluating a Definite Integral

Evaluate the definite integral \(\int_{\sqrt{3} / 3}^{\sqrt{3}} \frac{d x}{1 + x^{2}} .\)

Solution

Use the formula for the inverse tangent. We have

\[\begin{aligned} \\ \\ \int_{\sqrt{3} / 3}^{\sqrt{3}} \frac{d x}{1 + x^{2}} & = \text{tan}^{−1} x \left|\right._{\sqrt{3} / 3}^{\sqrt{3}} \\ & = \left[\right. \text{tan}^{−1} \left(\right. \sqrt{3} \left.\right) \left]\right. - \left[\right. \text{tan}^{−1} \left(\right. \frac{\sqrt{3}}{3} \left.\right) \left]\right. \\ & = \frac{\pi}{6} . \end{aligned}\]

Checkpoint 5.44

Evaluate the definite integral \(\int_{0}^{2} \frac{d x}{4 + x^{2}} .\)

This lesson is part of:

Integration

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