Evaluate Integral Using Trigonometric Substitution2345

Evaluate the integral $\displaystyle \int_0^{\frac{3\sqrt{3}}{2}} \frac{x^2}{(4x^2 + 9)^{3/2}} \, dx$ using trigonometric substitution and solve it as a definite integral.

This problem involves evaluating a definite integral using the method of trigonometric substitution, a common technique in calculus. Trigonometric substitution is often employed when dealing with integrals that contain expressions such as $a^2 + x^2$. The key idea is to simplify the integrand by substituting a trigonometric function for $x$, transforming the integral into a trigonometric form that is easier to evaluate.

For this particular problem, you would identify the right trigonometric substitution that simplifies the expression $(4x^2 + 9)^{3/2}$. Typically, this involves substituting $x$ with a function of $\theta$ that transforms the algebraic expression into a trigonometric identity. After making the substitution, the integral will often involve simpler trigonometric functions, allowing you to use basic integration techniques.

The second important aspect is to evaluate the integral as a definite one, which means adjusting the limits of integration after substitution. This is a crucial step, as it ensures the final result corresponds to the original integral's limits. Through this approach, the trigonometric substitution not only simplifies the integration process but also reinforces the understanding of adjusting limits in definite integrals.

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