Solving $\int \frac{dx}{(x^2 + y^2 + z^2)^{\frac{3}{2}}}$ This was in an old exam in a physics for mathematicians class. I haven't had to deal with these kind of integrals for a while and can't think of a decent substitution. I asked my teacher about it and he mumbled a bit and told me to "google it". I've tried a few obvious ones but none of them seemed to work. Any help would be greatly appreciated.

**Classic Substitution Method**

Let $x=\sqrt{y^2+z^2}\tan \theta$ and thus $dx=\sqrt{y^2+z^2}\sec^2 \theta \,d\theta$. Then,

$$\begin{align} \int\frac{dx}{(x^2+y^2+z^2)^{3/2}}&=\frac{1}{y^2+z^2}\int\cos \theta \,d\theta\\\\\\\ &=\frac{1}{y^2+z^2}\sin \theta +C\\\\\\\ &=\frac{1}{y^2+z^2}\frac{x}{\sqrt{x^2+y^2+z^2}}+C \end{align}$$