Evaluate Integral Using Trigonometric Substitution2

Evaluate $\displaystyle \int \frac{1}{9 - x^2} \, dx$ using appropriate trigonometric substitution.

Trigonometric substitution is a powerful technique for evaluating integrals, especially those that involve radicals or expressions like the one in this problem: $1/\left(9-x^2\right)$. This technique leverages the fundamental Pythagorean identities to convert a potentially challenging algebraic integral into a more manageable trigonometric one. The underlying concept is to simplify the expression under the square root by linking it to a familiar trigonometric identity. This type of substitution is particularly useful when your integral resembles the forms of $a^2-x^2$, $a^2+x^2$, or $x^2-a^2$.

In this problem, since you are dealing with $9-x^2$ under the fraction, the key is to recognize this as a difference of squares fitting the form of $a^2-x^2$, which suggests a sine or cosine substitution given the Pythagorean identity $\sin^2 + \cos^2 = 1$. By making an appropriate substitution, you can simplify the integral by transforming it into one involving trigonometric functions, which are often easier to integrate. This process not only aids in finding the integral but also provides deeper insight into the connections between algebraic and trigonometric functions.

Overall, mastering trigonometric substitution enhances your problem-solving toolkit for tackling a variety of integrals. It reinforces concepts of trigonometric identities and their applications, promoting an intuitive understanding of integration techniques, and demonstrates the versatility and interconnectedness of mathematics.

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