7.2 definite integral — ગણિત ધોરણ 12 સાયન્સ — Question

Put ${x^2} = \sin \theta $ $ \Rightarrow 2x\,dx = \cos \theta \,d\theta $

$I = \frac{1}{2}\int_0^{\pi /2} {\frac{{{{\sin }^3}\theta .\cos \theta \,\,d\theta }}{{\cos \theta }}} $

$= \frac{1}{2}\int_0^{\pi /2} {{{\sin }^3}\theta \,\,d\theta } $

$ = \frac{1}{2}\frac{{\Gamma 2\,\Gamma (1/2)}}{{2.\Gamma (5/2)}} $

$= \frac{{\Gamma \left( {\frac{1}{2}} \right)}}{{4.\frac{3}{2}.\frac{1}{2}.\Gamma \left( {\frac{1}{2}} \right)}} = \frac{1}{3}$.