Question

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

Answer 1

$\begin{array}{c}I=\underset{0}{\overset{\frac{\pi }{2}}{\int }}{\sin }^{2}x{\cos }^{3}xdx\\ I=\underset{0}{\overset{\frac{\pi }{2}}{\int }}{\sin }^{2}x{\cos }^{2}x\cos xdx\\ I=\underset{0}{\overset{\frac{\pi }{2}}{\int }}{\sin }^{2}x\left(1-{\sin }^{2}x\right)\cos xdx\\ \left\{\begin{array}{c}Let,\sin x=t;\cos xdx=dt\\ x=0;t=0\\ x=\frac{\pi }{2};t=1\end{array}\right\}\\ I=\underset{0}{\overset{1}{\int }}{t}^2\left(1-t^2\right)dt\\ I={\left[\frac{{\sin }^{3}x}{3}-\frac{{\sin }^{5}x}{5}\right]}_0^1=\frac{1}{3}-\frac{1}{5}=\frac{2}{15}\end{array}$