Question

Find the integral. $$ \int \frac{1}{\sqrt{x^{2}-9}} d x $$

   Find the integral.
$$
\int \frac{1}{\sqrt{x^{2}-9}} d x
$$

Calculus

Ron Larson, Robert… 8th Edition

Chapter 8, Problem 27

Instant Answer

Solved by Expert Harshita Goel

10/01/2021

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In this case, $a=3$. Show more…

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Find the integral. $$ \int \frac{1}{\sqrt{x^{2}-9}} d x $$

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Key Concepts

Trigonometric Substitution

This technique involves replacing a variable with a trigonometric function to simplify an integral containing square roots of quadratic expressions. In the context of integrals such as those involving ?(x² - a²), one uses a substitution like x = a sec? to transform the radical into a form involving tangent functions, making the integral easier to evaluate.

Standard Integral Formula for 1/?(x² - a²)

There exists a well-known standard formula in integral calculus: ? dx/?(x² - a²) = ln|x + ?(x² - a²)| + C. Recognizing when an integral fits this form allows for a direct application of the formula, thereby simplifying the integration process without the need for extensive substitution steps.

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