Simplify and Integrate Expression Using Trigonometric Substitution

Simplify and integrate the expression $(x^2 + 9)^{3/2}$ using trigonometric substitution where $x = 3\tan(\theta)$.

In this problem, we are tasked with simplifying and integrating the expression (x^2 + 9) raised to the power of 3/2 using a technique known as trigonometric substitution. This is an important method in integral calculus, particularly useful when dealing with integrals that contain square roots of quadratic expressions. Trigonometric substitution leverages the identities and properties of trigonometric functions to transform a difficult integral into a simpler form.

When we use the substitution $x = 3\tan(\theta)$, we are essentially replacing x with a trigonometric function. This substitution makes use of the identity $\tan^2(\theta) + 1 = \sec^2(\theta)$, which simplifies the expression under the square root. By substituting x with $3\tan(\theta)$, the expression under the square root becomes 9 sec^2(theta), which then simplifies to a constant factor times the integral of a power of secant.

After performing the substitution, the integration becomes more straightforward as it often involves an integral that is easier to solve, such as an integral of a power of a secant function. The process also includes transforming back from theta to x once the integration is complete by using the inverse trigonometric functions. Understanding this method not only aids in integrating complex functions but also reinforces knowledge of integration techniques and trigonometric identities.

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