Integrating x Cubed Over Square Root of 16 Minus x Squared Using Trigonometric Substitution

Integrate $\frac{x^3}{\sqrt{16-x^2}}$ using trigonometric substitution.

When tackling an integration problem involving expressions such as the square root of a difference of squares, a powerful technique that often comes into play is trigonometric substitution. This approach is particularly useful in simplifying integrals with square roots involving quadratic expressions. In this problem, you are prompted to handle an integral where the denominator has a square root of a difference of squares, specifically under the form of 16 minus x squared.

Trigonometric substitution works by leveraging the Pythagorean identities and the geometric representations of trigonometric functions. In this context, substituting x with a trigonometric expression like 4 sine theta or 4 cosine theta can transform the integrand into a trigonometric integral that is often simpler to solve. This substitution varies depending on the form of the expression under the square root, intentionally chosen to collapse the root into a more manageable format using the identity such as sine squared theta plus cosine squared theta equals one.

Once the substitution is made, the problem becomes one of integrating trigonometric functions, which are well-documented and systematically structured, enabling us to apply standard integration techniques. After performing the integration with respect to theta, it is crucial to convert the result back into the original variable x. This involves back-substitution using the initial trigonometric relationships established. Understanding how to make these substitutions and transformations effectively is a central component of mastering integration techniques involving trigonometric substitution.

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