Evaluating an Integral Using Trigonometric Substitution

Using the substitution $x = \sqrt{3} \sec(\theta)$, evaluate the integral $\displaystyle \int \frac{\sqrt{x^2 - 3}}{x} \, dx$.

Trigonometric substitution is a powerful technique in calculus used to simplify integrals involving square roots of quadratic expressions. In this problem, the substitution involves transforming the variable x using a trigonometric identity, specifically utilizing the secant function. The essence of this method lies in transforming the integral into a form that is easier to evaluate by using identities that simplify the square root expression.

When approaching integrals like this one, it's crucial to recognize patterns that match trigonometric identities. For instance, the presence of a square root of the form of x squared minus a constant suggests a substitution involving tangent or secant, which helps in transforming the expression under the integral into a more manageable form. Once the substitution is made, the integral often converts into a more standard form that involves basic trigonometric integrals.

After performing the substitution and changing the variable of integration, it is essential to also adjust the differentials and limits if they are present. The process oftentimes also requires reverting back to the original variable once the integration is complete, which involves using the inverse trigonometric functions to express the final result in terms of the original variables. This integrative method not only simplifies complex expressions but also enhances understanding of the interplay between algebraic and trigonometric functions.

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