Integral with Trigonometric Substitution Involving Square Root

Evaluate the integral $\displaystyle \int \frac{x}{\sqrt{x^2 - 7}} \, dx$ using trigonometric substitution.

To evaluate the given integral using trigonometric substitution, we first identify a suitable substitution that simplifies the expression under the square root. The integrand contains a square root of the form $x^2 - a^2$, which suggests employing the trigonometric substitution $x = a \sec(\theta)$, where $a = \sqrt{7}$. This substitution transforms the integrand into a more manageable form, enabling the integration process to proceed more smoothly. Through this approach, the differential $dx$ is replaced with $a \sec(\theta) \tan(\theta) d\theta$, while the square root simplifies to $a \tan(\theta)$. Transforming the integral into the trigonometric domain typically results in an integral involving basic trigonometric functions, which are often easier to integrate, especially if they reduce to a standard form after substitution. Once the integral is evaluated, it is crucial to remember the final step: converting back from the trigonometric variable to the original variable x using the inverse of the initial substitution. This step ensures that the solution is presented in terms of the original variable provided in the problem statement. Trigonometric substitution is a powerful method in calculus used for integrating expressions containing roots of quadratic polynomials. Its effectiveness lies in how it converts polynomial radicals into simple trigonometric functions, thereby leveraging the fundamental identities of trigonometry for integration. Understanding and mastering trigonometric substitution not only aids in evaluating integrals like the one presented here but also enhances problem-solving skills for a variety of integration challenges encountered in calculus.

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