Double integral with change of variables formula Say I want to find an expression for the integral $\int_{-\infty}^{\infty}\int_{-\infty}^{\infty}\frac{1}{(1+x^2+y^2)^c}dxdy$ for some $c$ using change of variables formula. I can't seem to identify any good change of variables. Trying converting to polar made it a bit of a mess, an in particular I'm unsure of how to deal with the infinite bounds on the integrals. Help would be appreciated.
 A: Change of variables is obviously a good strategy, but it might pretty quickly lead you to unfamiliar integrals with hyperbolic trigonometric functions (but do give it a try! The main idea is the derivative of $\arctan(x)$).
Since you mentioned polar coordinates, here's a push towards a solution using that. Note that your area of integration is the whole plan $\mathbb R^2$. You are basically breaking down your plane as a grid and then integrating along the grid lines. When you work with polar coordinates, you would break down your plane as circles centered at the origin. Here's how it goes.
\begin{align*}I=\int_{x=-\infty}^\infty\int_{y=-\infty}^\infty\frac{1}{(1+x^2+y^2)^c}\ \mathrm dy\mathrm dx &= \int_{\theta=0}^{2\pi}\int_{r=0}^\infty \frac{1}{(1+r^2)^c}\ r\mathrm dr\mathrm d\theta\\&=\int_{\theta=0}^{2\pi}\int_{r=0}^\infty
\frac{r\mathrm dr}{(1+r^2)^c}\ \mathrm d\theta\end{align*}
Hopefully you can take it from here. So that you can check your answer after you are done, here's the solution in spoiler tags.

 $I=\frac{\pi}{c-1}$ for $c>1$. For $c\le 1$, the integral does not converge.

