Question

Evaluate $\int {\sin }^{3}x\sin 2xdx$.

Answer 1

$\int {\sin }^{3}x\sin 2xdx$

$=\int {\sin }^{3}x2\sin x\cos xdx$

$I=2\int {\sin }^{4}x\cos xdx$

Let $t=\sin x\Rightarrow dt=\cos xdx$

$I=2\int t^{4}dt$

$=2\left(\frac{t^5}{5}\right)+c$ , where $c$ is the constant of integration

$=\frac{2{\sin }^{5}x}{5}+c$ , where $t=\sin x$