For every natural integer $k$ we set $$m_{k} = \frac{1}{2\pi} \int_{-2}^{2} x^{k} \sqrt{4 - x^{2}} \, \mathrm{d}x$$ Using integration by parts, show that, for every natural integer $k$, $$m_{2k+2} = \frac{2(2k+1)}{k+2} m_{2k}$$
For every natural integer $k$ we set
$$m_{k} = \frac{1}{2\pi} \int_{-2}^{2} x^{k} \sqrt{4 - x^{2}} \, \mathrm{d}x$$
Using integration by parts, show that, for every natural integer $k$,
$$m_{2k+2} = \frac{2(2k+1)}{k+2} m_{2k}$$