Chapter 11.7, Problem 4P

Express the following integrals as $B$ functions, and then, by $(7.1),$ in terms of $\Gamma$ functions. When possible, use $\Gamma$ function formulas to write an exact answer in terms of $\pi ,\sqrt{2},$ etc. Compare your answers with computer results and reconcile any discrepancies.

${\displaystyle {\int }_0^1{x}^2}{\left(1-x^2\right)}^{3/2}dx$