Question

Solve : $2\underset{0}{\stackrel{\frac{\pi }{2}}{\int }}{\sin }^{2}x{\cos }^{3}xdx$

Answer 1

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

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

$=\int {\sin }^{2}x(1-{\sin }^{2}x)\cos xdx$

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

$u=\sin x\Rightarrow (\sin x=0\to u=0,\sin x=\frac{\pi }{2}\to u=1)du=\cos x$

$=\int (u^2-u^4)du$

$={[\frac{u^3}{3}-\frac{u^5}{5}]}_0^1$

$=[\frac{1}{3}-\frac{1}{5}]$

$=\frac{2}{15}$

$2\int {\sin }^{2}x{\cos }^{3}xdx=\frac{4}{15}$