<s>[INST] Is there a nice way to integrate this? I'm trying to integrate:
$$\iiint\limits_{x^2+y^2+z^2\leqslant 1}x^{2n}+y^{2n}+z^{2n}\mathrm{d}V$$
but, I can't seem to find any nice way to do this?
 [/INST] $$\begin{align}
  & \int_{x^2+y^2+z^2\le 1} x^{2n}+y^{2n}+z^{2n} dxdydz\\ 
= & 3 \int_{x^2+y^2+z^2\le 1}x^{2n} dx dy dz\\
= & 3 \int_{-1}^{1} dx\left( x^{2n} \int_{y^2 + z^2 \le 1 - x^2} dy dz \right)\\
= & 3\pi \int_{-1}^{1} dx \left( x^{2n} (1 - x^2)\right)\\
= & 3 \pi \left( \frac{2}{2n+1} - \frac{2}{2n+3} \right)\\
= & \frac{12 \pi}{(2n+1)(2n+3)}
\end{align}$$
</s>