Integral with Trigonometric Substitution Involving Square Root2

Evaluate the integral $\displaystyle \int \frac{\sqrt{x^2 - 16}}{x^2} \, dx$ using trigonometric substitution.

Trigonometric substitution is a powerful technique for evaluating integrals, particularly when dealing with expressions involving square roots of quadratic forms. In this type of problem, we use substitutions based on trigonometric identities to simplify the integral into a more workable form. This approach is especially useful when the integrand contains expressions like the square root of a sum or difference of squares. The first step involves identifying the appropriate trigonometric substitution. For expressions like the square root of x squared minus a constant, substituting x with a trigonometric function, such as a secant or tangent, can transform the integrand into a rational function of the trigonometric variable. This simplification often leads to a form that is easier to integrate. Once the substitution is made, the integral can frequently be tackled using standard techniques for integrating trigonometric functions or simpler algebraic manipulations. After integrating, it is crucial to convert back to the original variable. Since we originally substituted using a trigonometric identity, we will use inverse trigonometric functions to express the solution in terms of the original variable. This process requires careful attention to algebraic manipulation and understanding of trigonometric identities, yet offers a systematic way to handle integrals involving square roots of quadratic expressions.

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