Question

The value of ${\int }_{-\pi }^{\pi }{\sin }^{3}x{\cos }^{2}x\mathrm{dx}$ is equal to

Answer 1

$\mathrm{Let}f\left(x\right)={\sin }^{3}x{\cos }^{2}x\therefore f\left(-x\right)=-{\sin }^{3}x{\cos }^{2}x\mathrm{Hence},f\left(x\right)\mathrm{is}\mathrm{an}\mathrm{odd}\mathrm{function}.\mathrm{We}\mathrm{know},{\int }_{-a}^{a}f\left(x\right)\mathrm{dx}=0\mathrm{if}f\left(x\right)\mathrm{is}\mathrm{an}\mathrm{odd}\mathrm{function}.\therefore {\int }_{-\pi }^{\pi }{\sin }^{3}x{\cos }^{2}x\mathrm{dx}=0$