Integrate x cubed over square root of one minus x squared

Integrate the integral of $\frac{x^3}{\sqrt{1-x^2}}$ with respect to $x$.

When faced with the integral of x cubed over the square root of one minus x squared, one strategy to use is substitution. Trigonometric substitution is particularly useful when dealing with integrals involving square roots of expressions like one minus x squared. This is because the trigonometric identity cosine squared plus sine squared equals one suggests that expressions involving squares can often be related to trigonometric functions. By using a trigonometric substitution, such as substituting x with sine theta, the integral can be transformed into an expression that is simpler to integrate in terms of theta. This approach not only simplifies the square root but often results in standard trigonometric integrals that are easier to solve.

Beyond substitution, it's essential to consider the structure of the integrand and how it relates to potential trigonometric identities. The derivative of a trigonometric substitution often transforms the integral into a form that is more straightforward. Additionally, evaluating the boundaries or limits of the original x values in terms of theta after substitution can help ensure that the integration is carried out correctly, particularly if the original problem specifies limits of integration. It is a prime example of how a seemingly complex rational algebraic problem can become manageable through the appropriate use of trigonometric identity techniques.

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