Integral of x cubed times square root of 9 minus x squared

Evaluate the integral $\displaystyle \int x^3 \sqrt{9 - x^2} \, dx$ using trigonometric substitution.

This problem focuses on evaluating an integral using trigonometric substitution. Trigonometric substitution is a technique often used to simplify integrals involving square roots of quadratic expressions. By substituting a trigonometric function for a variable, the integral can be transformed into a trigonometric integral, which is often easier to evaluate. In this specific case, the expression involves a square root of the form square root of a constant minus a variable squared, which suggests using the sine or cosine substitution due to their Pythagorean identities. Such substitutions transform the integral into a trigonometric form, which can then be integrated using standard techniques. The use of trigonometric identities can simplify these expressions significantly.

Understanding when to apply trigonometric substitution involves recognizing the form of the integral and identifying the appropriate trigonometric identity to use. After substitution, the problem often reduces to a more manageable form that can be integrated directly. One must also be careful to convert the limits of integration if the original integral was definite, and to substitute back to the original variable to find the antiderivative if it was indefinite. This approach is a powerful tool in solving integrals that appear complex due to the presence of square roots of quadratic expressions.

Related Problems