Is the function $f(x,y) = \frac{x^2 - y^2}{(x^2+y^2)^2}$ lebesgue integrable in $[0,1]\times[0,1]$?

Question

So, there's something that I can use to show that $f(x,y) = \frac{x^2 - y^2}{(x^2+y^2)^2}$ is or is not Lebesgue integrable in $[0,1]\times[0,1]$?

Answer 1

Hint: Consider the integral over $0\le r \le 1, 0\le \theta \le \pi/8,$ in polar coordinates.

Answer 2

The only problem is at the origin, no? Perhaps you might try to isolate this bad point? Say, $B_0=B(0,\epsilon)\bigcap[0,1]^2$, and $B_0^C=[0,1]^2-B_0$. Then we need only consider the integral $\int_{B_0}fdx$ to investigate integrability.

Also, by the form of the integrand (only squared's show up), what does the integrand look like in polar coordinates?

$$\int_{B_0}fdx=\int_0^{\frac{\pi}{2}}\int_0^\epsilon\frac{\cos(2\theta)}{r}drd\theta.$$ Do we expect this integral to be finite?