Integrable or not ???

Question

How to check whether a function is integrable or not: $f(x,y)=\frac{1}{1+x^2+y^2}$ over $\Bbb{R}^2$.

I am not able to start the problem. Is there some theorem which deals with such problems?

Answer 1

Using polar coordinates, as recommended in the comments:

$$\begin{cases}x=r\cos\theta\\{}\\y=r\sin\theta\end{cases}\;\;,\;\;0\le r<\infty\;,\;\;0\le\theta\le 2\pi$$

so the wanted integral is (don't forget the Jacobian):

$$\int_0^\infty\int_0^{2\pi}\frac r{1+r^2}\, d\theta\,dr=\left.2\pi\int_0^\infty\frac r{1+r^2}dr=\pi\log(1+r^2)\right|_0^\infty$$

and since the above limit is infinite the integral doesn't exist (finitely).