Question Number 17525 by alex041103 last updated on 07/Jul/17
$$\mathrm{Evaluate}\:\underset{−\mathrm{1}} {\overset{\mathrm{1}} {\int}}\left(\mathrm{1}−{x}^{\mathrm{2}} \right)^{\frac{{n}}{\mathrm{2}}} {dx}\:\:\mathrm{for}\: \\ $$$${n}\:\in\:\mathbb{Z}\cap\left[\mathrm{0};\infty\right)\:\left(\mathrm{i}.\mathrm{e}.\:\mathrm{0},\:\mathrm{1},\:\mathrm{2},\:…\right)\:\mathrm{and}: \\ $$$$\left.\boldsymbol{\mathrm{a}}\right)\:\:{n}\:\equiv\:\mathrm{0}\left({mod}\:\mathrm{2}\right) \\ $$$$\left.\boldsymbol{{b}}\right)\:{n}\:\equiv\:\mathrm{1}\left({mod}\:\mathrm{2}\right) \\ $$
Commented by alex041103 last updated on 07/Jul/17
$$\mathrm{This}\:\mathrm{is}\:\mathrm{not}\:\mathrm{from}\:\mathrm{a}\:\mathrm{book}! \\ $$