Question

Evaluate ${\int }_0^{2\pi }{\sin }^{100}x{\cos }^{99}xdx$

Answer 1

$I={\int }_0^{2\pi }{\sin }^{100}x{\cos }^{99}xdx.........\left(i\right)I={\int }_0^{2\pi }{\sin }^{100}(2\pi -x){\cos }^{99}(2\pi -x)dxI=-{\int }_0^{2\pi }{\sin }^{100}x{\cos }^{99}xdx.......\left(ii\right)$

Adding $\left(i\right)$ and $\left(ii\right)$

$2I=0$

$I=0$