Evaluate Definite Integral with Trigonometric Substitution

Evaluate the definite integral $\displaystyle \int_0^{\frac{3}{2}} \frac{1}{\sqrt{9-x^2}} \, dx$ using trigonometric substitution.

In this problem, we delve into the technique of evaluating definite integrals using trigonometric substitution, a powerful method particularly useful when dealing with integrands containing expressions like the square root of a difference squared. The integral given, featuring the expression nine minus x squared under the square root, is a classic candidate for such a substitution. By employing trigonometric identities, we can simplify the integrand into a form that is easier to integrate—typically transforming it into a trigonometric integral that can be resolved using basic trigonometric identities or integral tables.

The process generally involves a substitution that transforms the variable x into a trigonometric function of a new variable, theta, which simplifies the square root in the denominator. This substitution often leads to an integral in terms of theta that is straightforward to evaluate. Once the integral is evaluated in terms of the trigonometric variable, it’s necessary to convert back into terms of the original variable using the inverse substitution. This approach not only aids in solving the problem at hand but also enhances understanding of how trigonometric functions can be used to simplify complex algebraic expressions, a fundamental concept in calculus. Mastery of this technique extends to various applications, including physics and engineering problems where such expressions frequently arise.

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