Integrate One Over X Times Square Root of Nine Minus X Squared

Integrate $\displaystyle \frac{1}{x \sqrt{9-x^2}}$.

This problem involves integrating a function with a square root in the denominator, combined with a variable term, often associated with trigonometric substitution methods. The integral of one over x times the square root of nine minus x squared can be approached effectively by recognizing patterns that are similar to inverse trigonometric function derivatives.

When dealing with expressions of the form of a square root involving subtraction of a squared term, such as the square root of nine minus x squared, trigonometric substitution is a powerful tool. The substitution often employs related trigonometric identities to simplify the square root expression. In this problem, one might consider substituting x with a trigonometric function like sine or cosine, exploiting the identity that relates one minus sine squared to cosine squared, or vice versa. This can simplify the expression under the square root and may lead to an integration involving basic trigonometric forms.

Furthermore, understanding the geometric interpretation of the expression can be insightful. The expression nine minus x squared under a square root is reminiscent of the equation of a circle. Therefore, through trigonometric substitution, you are effectively mapping the problem onto a segment of a circle, easing the integration process as the trigonometric identities align with circular geometry. This abstract thinking aids in transforming what seems a complex algebraic function into a more straightforward trigonometric integral.

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