Integral with Trigonometric Substitution

Evaluate the integral of the form $\displaystyle \int \frac{dx}{\sqrt{x^2 - a^2}}$ by making the substitution $x = a \sec\theta$.

Trigonometric substitution is a powerful technique for evaluating integrals that involve square roots of quadratic expressions. By substituting a trigonometric function for the variable, the integral can be transformed into a more manageable form. This particular problem involves the substitution $x = a \sec\theta$, which simplifies the square root expression into one involving tangent, allowing for easier integration.

The underlying idea is to utilize trigonometric identities to transform the integral into a simpler form, leveraging the fact that trigonometric functions often have straightforward derivatives and integrals. In this case, the substitution helps in simplifying the complex radical expression which otherwise might be cumbersome to integrate directly.

This technique is particularly useful when dealing with integrals involving radicals like square roots or even higher roots, where direct integration is not feasible. The choice of substitution often depends on the specific structure of the integral, and recognizing these forms is crucial for efficiently solving such problems. Students should focus on understanding the types of substitutions that correspond with various integrals to develop a versatile integration strategy.

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