Integration Using Trigonometric Substitution2

Integrate $\frac{y^2}{(16-y^2)^{3/2}}$ with respect to $y$.

Integration problems that involve radicals often require creative techniques for simplification, and one such technique is trigonometric substitution. In this problem, the integral of $y^2/(16-y^2)^{3/2}$ is best approached by recognizing the expression under the square root as a difference of squares, resembling the Pythagorean identity of sine and cosine. By substituting $y = 4\sin(\theta)$, we simplify the expression $(16-y^2)$ into a trigonometric form where $\cos(\theta)$ can be easily integrated.

This method works because trigonometric identities provide a path to remove the square root, thereby reducing the complexity of the integration. In this specific case, the substitution leads to a change of variables that transforms the integral to a more straightforward form involving trigonometric functions. After performing the substitution and simplifying, the resulting integral often involves basic trigonometric identities and derivatives, which are more manageable.

Trigonometric substitution is a powerful tool in integral calculus, especially when dealing with radicals or quadratic expressions that match the form of trig identities like $a^2 - x^2$, $a^2 + x^2$, or $x^2 - a^2$. Learning to identify these patterns will also aid in other integration techniques, such as integration by parts or partial fractions, as many integrals in calculus require multiple techniques to evaluate efficiently. This problem serves as a good practice for recognizing when and how to apply trigonometric identities in integration, illustrating not only the procedural steps but also the strategic thinking involved in tackling integrals of this nature.

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