Simplify Integral Using Trigonometric Substitution

Simplify the integral $\displaystyle \int \frac{1}{\sqrt{9 - x^2}} \, dx$ using the substitution $x = 3 \sin \theta$.

Trigonometric substitution is an invaluable technique, especially for integrals involving square roots of quadratic expressions. In this problem, you're dealing with an integral where the function within the square root is similar to a Pythagorean identity. By using the substitution x equals 3 times sine of theta, you transform the integral into a form that is easier to handle. This substitution is effective because it leverages the identity for sine: sine squared theta plus cosine squared theta equals 1, which simplifies the expression under the square root to involve only cosine.

Once you perform the substitution, this integral simplifies to a form involving a trigonometric function whose integral is easy to evaluate. This highlights the strategy behind trigonometric substitution: convert a difficult algebraic expression into a trigonometric one, solve the simplified problem, and then revert the substitution. Mastering this form of substitution requires practice in recognizing the types of quadratic expressions that benefit from trigonometric identities.

Understanding when and how to apply trigonometric substitution not only simplifies particular types of integrals but also enriches your toolkit for a broader range of calculus problems. It deepens your appreciation for the interconnectedness of algebra and trigonometry in calculus, which is particularly useful in solving complex problems efficiently.

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