Consider the improper integral
J=∫011−x6x2dx,
which arises in the normalization of a probability density function.
(a) Evaluate the indefinite integral ∫1−x6x2dx and express the antiderivative in elementary functions.
(b) Hence determine the exact value of J. 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.