Integrate functions resulting in inverse trigonometric functions
Integrals that Result in Inverse Sine Functions
Let us begin with the three formulas. Along with these formulas, we use substitution to evaluate the integrals. We prove the formula for the inverse sine integral.
Integration Formulas Resulting in Inverse Trigonometric Functions
The following integration formulas yield inverse trigonometric functions:
Evaluating a Definite Integral Using Inverse Trigonometric Functions
Evaluate the definite integral [latex]{\displaystyle\int }_{0}^{\frac{1}{2}}\dfrac{dx}{\sqrt{1-{x}^{2}}}.[/latex]
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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
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Find the indefinite integral using an inverse trigonometric function and substitution for [latex]\displaystyle\int \frac{dx}{\sqrt{9-{x}^{2}}}.[/latex]
Hint
Use the formula in the rule on integration formulas resulting in inverse trigonometric functions.
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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: Finding an Antiderivative Involving the Inverse Tangent Function
Find an antiderivative of [latex]\displaystyle\int \frac{1}{1+4{x}^{2}}dx.[/latex]
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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 [latex]{ \tan }^{-1}u+C.[/latex] So we use substitution, letting [latex]u=2x,[/latex] then [latex]du=2dx[/latex] and [latex]\frac{1}{2}du=dx.[/latex] Then, we have
Use the solving strategy from finding an antiderivative involving an inverse trigonometric function and the rule on integration formulas resulting in inverse trigonometric functions.
Example: Applying the Integration Formulas
Find the antiderivative of [latex]\displaystyle\int \frac{1}{9+{x}^{2}}dx.[/latex]
Evaluate the definite integral [latex]{\displaystyle\int }_{0}^{2}\dfrac{dx}{4+{x}^{2}}.[/latex]
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[latex]\frac{\pi }{8}[/latex]
Hint
Follow the procedures from (Figure) to solve the problem.
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