7.2 definite integral — Maths STD 12 Science — Question

Put $x = a\sin \theta \Rightarrow dx = a\cos \theta \,d\theta $

$I = \int_0^{\pi /2} {{a^2}{{\sin }^2}\theta .{a^3}{{\cos }^3}\theta .a\cos \theta \,d\theta } $

$ = {a^6}\int_0^{\pi /2} {{{\sin }^2}\theta {{\cos }^4}\theta \,d\theta= {a^6}\frac{{\Gamma \frac{3}{2}.\,\Gamma \frac{5}{2}}}{{2.\Gamma \frac{8}{2}}}} $

$= {a^6}\frac{{\frac{1}{2}.\sqrt \pi .\frac{3}{2}.\frac{1}{2}.\sqrt \pi }}{{2.3.2.1}} $

$= \frac{{\pi {a^6}}}{{32}}$.