7.2 definite integral — ગણિત ધોરણ 12 સાયન્સ — Question

$ = 2\int_0^{^{\pi /2}} {{{\sin }^4}x\,{{\cos }^6}x.\,dx} $

$\begin{matrix}
   \because \int_{-a}^{a}{f(x)\,dx=2\int_{0}^{a}{f(x)\,dx,}} & \text{if }f(-x)=f(x)  \\
   \,\,\,\,\,=0, & \text{if }f(-x)=-f(x)  \\
\end{matrix}$

Applying Gamma function, we get

$I = \frac{{2\,\Gamma 5/2\,.\,\Gamma 7/2}}{{2\,.\Gamma 6}}$

$ = \frac{{3/2.1/2.\sqrt {\pi .} 5/2.3/2.1/2.\sqrt \pi }}{{5.4.3.2.1}}$

$ = \frac{{3\pi }}{{{2^8}}} = \frac{{3\pi }}{{256}}$.