grandes-ecoles 2021 Q17

grandes-ecoles · France · centrale-maths1__mp Indefinite & Definite Integrals Definite Integral Evaluation (Computational)
For every natural integer $k$ we set $$m_{k} = \frac{1}{2\pi} \int_{-2}^{2} x^{k} \sqrt{4 - x^{2}} \, \mathrm{d}x$$ For $k \in \mathbb{N}$, what is the value of $m_{2k+1}$?
For every natural integer $k$ we set
$$m_{k} = \frac{1}{2\pi} \int_{-2}^{2} x^{k} \sqrt{4 - x^{2}} \, \mathrm{d}x$$
For $k \in \mathbb{N}$, what is the value of $m_{2k+1}$?