Integral of Square Root of Nine Minus X Squared Over X Squared Using Trig Substitution

Evaluate the integral $\displaystyle \int \frac{\sqrt{9 - x^2}}{x^2} \, dx$ using trig substitution, where you substitute $x = 3\sin\theta$.

In this problem, we approach the integration of a function through the method of trigonometric substitution. Trigonometric substitution is a powerful technique to simplify integrals involving radicals, particularly those of the form involving square roots of quadratic polynomials. By substituting x with a trigonometric expression, such as $x = 3 \sin(\theta)$ in this case, we leverage the Pythagorean identity to transform the integral into a trigonometric integral that is often easier to solve.

The choice of substitution depends on the type of radical expression. For the square root of a squared minus x squared, substituting x with a times sine of theta is particularly useful, as it simplifies the square root term using the identity $1 - \sin^2(\theta) = \cos^2(\theta)$, which can be easily integrated with respect to theta. This process generally requires transforming the differential, adjusting integrals, and sometimes using additional trigonometric identities or algebraic manipulation to simplify the result.

Understanding the concept and application of trigonometric substitution helps students solve a broad range of integration problems more efficiently. It also reinforces the connection between algebraic and trigonometric techniques, which are fundamental in calculus.

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