Explanation

(c)
$I = \int_{- \pi/2}^{\pi/2}{\sin^{4}{x\cos^{6}{x\ dx}}}$
This is an even function.
$\therefore I = 2\int_{0}^{\pi/2}{\sin^{4}{x\cos^{6}{x\ dx}}}$
Applying Gamma function
$\Rightarrow \ \ I = \frac{\sqrt{\frac{4 + 1}{2}} - \sqrt{\frac{6 + 1}{2}}}{2|6} = \frac{2\sqrt{\frac{5}{2}} - \sqrt{\frac{7}{2}}}{2|6}$
$= \frac{\frac{3}{2} \bullet \frac{1}{2} \bullet \sqrt{\pi} \bullet \frac{5}{2} \bullet \frac{3}{2} \bullet \frac{1}{2} \bullet \sqrt{\pi}}{5 \bullet 4 \bullet 3 \bullet 2 \bullet 1} = \frac{3\pi}{2^{8}} = \frac{3\pi}{256}$