$I= \int \sin ^3 x \sin 2x dx$ Find the value of I

Question

\[\begin {array} {1 1} (a)\;\frac{1}{5} \\ (b)\;\frac{2}{5} \\ (c)\;\frac{3}{5} \\ (d)\;0 \end {array}\]

Answer 1

$\int \limits_0^{\frac{\pi}{2}} \sin ^3 x \sin 2x dx$

$\qquad= \int \limits_0^{\frac{\pi}{2}} 2 \sin ^4 x \cos x dx$

$\sin x =t$

$\cos x dx=dt$

$\int \limits _0^1 2 t^4 dt$

$\qquad= \large\frac{2t^5}{5} \bigg]_0^1 $

$\qquad=\large\frac {2}{5}$

Hence b is the correct answer.