Integral with Trigonometric Substitution and Square Root

Evaluate the integral $\displaystyle \int \frac{\sqrt{9-x^2}}{x^2} \, dx$ using trigonometric substitution.

In this problem, we explore the technique of trigonometric substitution, a powerful method used to evaluate certain integrals, especially those involving square roots that suggest the form of a trigonometric identity. Here, the integral involves the square root of a difference of squares, 9 - x^2, which is reminiscent of the identity associated with sine and cosine functions. By substituting x with a trigonometric function, we aim to simplify the expression under the integral, thereby converting it into an integral that is easier to evaluate.

The essence of trigonometric substitution in this context is to take advantage of the Pythagorean identity: $\sin^2(\theta) + \cos^2(\theta) = 1$. Substituting x as $3\sin(\theta)$ transforms the integrand into a form where the trigonometric identity can be applied, effectively turning the problem into evaluating an expression in terms of $\theta$. Afterwards, converting back to x will provide the solution to the original integral.

Understanding this technique not only helps in solving this particular problem but also provides insights into dealing with similar integrals across various applications in calculus. Recognizing when to use trigonometric substitution is crucial as it greatly simplifies the process of integration for problems involving certain algebraic structures.

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