Consider the improper integral

MATHEMATICS · Multi-part · AP · Calculus BC · Integration and Accumulation of Change · Integration and Accumulation of Change

Problem

Consider the improper integral J=∫01x21−x6 dxJ = \displaystyle \int_{0}^{1} \frac{x^{2}}{\sqrt{1-x^{6}}}\,dx, which arises in the normalization of a probability density function. (a) Evaluate the indefinite integral ∫x21−x6 dx\displaystyle \int \frac{x^{2}}{\sqrt{1-x^{6}}}\,dx and express the antiderivative in elementary functions. (b) Hence determine the exact value of JJ. State whether the integral converges absolutely. (c) Verify your result in part (b) by differentiating your antiderivative and commenting on the continuity of the integrand at the endpoints.

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