How to calculate the triple integral $\iiint_{\Omega_t}\frac{1}{(x^2+y^2+z^2)^\frac32}$?

Question

$\iiint_{\Omega_t}\frac{1}{(x^2+y^2+z^2)^\frac32}$, where $\Omega_t$ is the ellipsoid and $\Omega_t=\frac{x^2}{a^2}+\frac{y^2}{b^2}+\frac{z^2}{c^2}\leq 1$

I want to use the change variable for $u=\frac xa, v=\frac yb,w=\frac zc$, but after this, then we may have a complicated form for inside thing. How can I do for this integral. Should I think about the Gauss Theorem?

Answer 1

Hint

Do the substitution $$(x,y,z)=(ar\sin(\theta )\cos(\varphi ),br\sin(\theta )\sin(\varphi ),cr\cos \theta ),$$

where $\theta \in (0,\pi)$ and $\varphi \in (0,2\pi)$ and $r\in(0,1)$.