Prove that $f(x,y)=\frac{xy}{x^2+y^2}$ is not Lebesgue integrable on $A = [-1,1]\times [-1,1]$

Question

Prove that $f(x,y)=\frac{xy}{x^2+y^2}$ is not Lebesgue integrable on $A = [-1,1]\times [-1,1]$

To my knowledge I need to use Fubini's theorem. But this doesn't work because the integration would be 0. Can anyone give me some hints? Many thanks

Answer 1

If one uses polar coordinates, it seems that, as is any trigonometric polynomial, $f$ is integrable over any compact set: $$f(x,y)=f(r\cos u,r \sin u)=\cos u \sin u.$$