Evaluate Definite Integral Using Trigonometric Substitution in Two Ways

Evaluate the definite integral $\displaystyle \int_{4}^{6} \frac{x^2}{\sqrt{x^2 - 9}} \, dx$ by using trigonometric substitution in two different ways: first by computing the antiderivative and then using the Fundamental Theorem of Calculus with the original limits of integration; second by changing the limits of integration after substituting $x = 3 \sec(\theta)$ and evaluating the integral in terms of $\theta$.

This problem involves evaluating a definite integral using trigonometric substitution, and it challenges you to approach the problem in two distinct ways. Trigonometric substitution is a technique used to simplify integrals containing radical expressions. It leverages trigonometric identities to convert the integrand into a form easier to integrate. In this instance, substituting using trigonometric identities such as secant and tangent can help manage the square root in the denominator, which otherwise complicates direct integration.

In the first method, you compute the antiderivative of the function, transforming the problem into a task of integrating a trigonometric function. Understanding the properties of trigonometric functions and their derivatives is crucial here. Once the antiderivative is determined, the evaluation using the Fundamental Theorem of Calculus with the original limits provides the result of the definite integral. This method solidifies foundational concepts where integration builds upon derivative knowledge but in reverse.

The second method involves altering the limits of integration post substitution, a maneuver best understood through the connection of geometric interpretation of trigonometric substitution. The trigonometric substitutions map known intervals to simpler intervals for integration purposes. By directly changing variables from x to \( \theta \), and consequently modifying limits of integration, this method demonstrates a different aspect of calculus where transformations simplify the problem before integration. This alternative approach not only explores the symmetry and periodic nature of trigonometric functions but also strategically uses substitution to reflect this in the integration process. Such dual approaches illuminate the importance of flexibility in mathematical problem-solving, showcasing how different methods can be applied to solve the same problem efficiently.

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