$\int_{0}^{\infty}\int_{1}^{\infty}\frac{x^2-y^2}{(x^2+y^2)^2}dxdy$ diverges? I want someone to review my proof that $$\int_{1}^{\infty}\int_{0}^{\infty}\frac{x^2-y^2}{(x^2+y^2)^2}dxdy$$ does not converge.
To make things easier, I said let's look at the entire first quadrant and then subtract the integral over the small rectangle that we added.
move to polar coordinates:
$$\int_{0}^{\frac{\pi}{2}}\int_{o}^{\infty}r^2\frac{(\cos^2\theta-\sin^2\theta)}{r^3}drd\theta=\int_{0}^{\frac{\pi}{2}}\int_{o}^{\infty}\frac{\cos^2\theta-\sin^2\theta}{r}drd\theta=\int_{0}^{\frac{\pi}{2}}\int_{o}^{\infty}\frac{1-2\sin^2\theta}{r}drd\theta\leq\int_{0}^{\frac{\pi}{2}}\int_{o}^{\infty}\frac{1}{r}drd\theta$$
The inequality part is true since $-2\sin^2 x$ is always negative.
So:
$$\int_{0}^{\frac{\pi}{2}}\int_{o}^{\infty}\frac{1}{r}drd\theta=\frac{\pi}{2}\ln(r)|_0^{\infty}=\frac{\pi}{2}\ln(\frac{\infty}{0})=\frac{\pi}{2}\ln({\infty})=\infty$$
So over the entire first quadrant it diverges.
Is there a point to checking the small rectangle that we added? I mean, even if it diverges, the answer would still diverge.

As an iterated integral, this integral converges. Indeed
$$ \int_{0}^{\infty} \frac{x^{2} - y^{2}}{(x^{2} + y^{2})^{2}} \, dx = \left[ - \frac{x}{x^{2} + y^{2}} \right]_{0}^{\infty} = 0. $$
So the whole integral also converges to 0. But it does not converge absolutely. Indeed,
$$ \int_{0}^{\infty} \left|\frac{x^{2} - y^{2}}{(x^{2} + y^{2})^{2}}\right| \, dx = \int_{0}^{y} \frac{y^{2} - x^{2}}{(x^{2} + y^{2})^{2}} \, dx + \int_{y}^{\infty} \frac{x^{2} - y^{2}}{(x^{2} + y^{2})^{2}} \, dx = \frac{1}{y} $$
and therefore
$$ \int_{1}^{\infty} \int_{0}^{\infty} \left|\frac{x^{2} - y^{2}}{(x^{2} + y^{2})^{2}}\right| \, dxdy = \infty. $$
Much frustrating fact is that if you switch the order of integration, you get
$$ \int_{1}^{\infty} \frac{x^{2} - y^{2}}{(x^{2} + y^{2})^{2}} \, dy = \left[ \frac{y}{x^{2} + y^{2}} \right]_{1}^{\infty} = - \frac{1}{x^{2}+1} $$
and thus the whole inegral becomes
$$ \int_{0}^{\infty}\int_{1}^{\infty} \frac{x^{2} - y^{2}}{(x^{2} + y^{2})^{2}} \, dydx =  -\frac{\pi}{2}. $$
