Computation of an iterated integral I want to prove $$\int\limits_{-\infty}^\infty\int\limits_{-\infty}^\infty\frac{\sin(x^2+y^2)}{x^2+y^2}dxdy=\frac{\pi^2}{2}.$$ Since the function $(x,y)\mapsto\sin(x^2+y^2)/(x^2+y^2)$ is not integrable, I can't use the Theorem of Change of Variable. So, I'm trying to use residue formulae for some suitable holomorphic function to compute the inner integral, but I can't continue. Can someone suggest me a hint to solve this problem?
Addendum: I may be wrong, but I suspect Theorem of Change of Variable (TCV) is not the answer. The reason is the following: the number $\pi^2/2$ is gotten if we apply polar coordinates, but TCV guarantees that if we apply any other change of variable we can get the same number, $\pi^2/2$. If this function were integrable, this invariance property would be guaranteed, but it is not the case. Thus we may have strange solutions to this integral.

using polar coordinates we have:
$$
\lim_{R\rightarrow\infty}\int_{x^2+y^2 \le R^2} \frac{\sin(x^2+y^2)}{x^2+y^2}dxdy=\lim_{R\rightarrow\infty}\int_0^{R}\int_0^{2\pi}\frac{\sin (r^2)}{r^2}r d\theta dr=
$$
$$
=2\pi\int_0^{+\infty} \frac{\sin (r^2)}{r}dr=2\pi \frac{\pi}{4}=\frac{\pi^2}{2}
$$
