Solve multiple integral $\int_{-2}^{1} \int_{-2}^{1} \int_{-2}^{1} \frac{x^2}{x^2+y^2+z^2} \,dx \,dy \,dz$

Question

I found this integral in a book. I thought it was easy to solve it, but I've not been able to do it. I tried to use spherical coordinates, however, I'm not sure what the limits are in such coordinates. Any suggestion?

$$\int_{-2}^{1} \int_{-2}^{1} \int_{-2}^{1} \frac{x^2}{x^2+y^2+z^2} \,dx \,dy \,dz$$

Answer 1

Per the symmetry $$\int_{-1}^1 \int_{-1}^1 \int_{-1}^1 \frac{x^2}{x^2+y^2+z^2} \,dx \,dy \,dz\\ =\frac13 \int_{-1}^1 \int_{-1}^1 \int_{-1}^1 \frac{x^2+y^2+z^2}{x^2+y^2+z^2}\,dx\, dy \,dz =\frac83$$