Let $X=[-1,1],Y=[0,1]$, and $$f(x,y)=\dfrac{xy}{(x^2+y^2)^2}$$ for $x\in X,y\in Y$. Let $\mu$ be the Lebesgue measure. Does the following integral exist:
$$\int_{X\times Y} f(x,y)d(\mu\times\mu)$$
It seems very hard to follow the definition of Lebesgue integrals using simple functions here. To do that we must find simple functions that are lower bounds for the function $f$. But the form of $f$ is quite complicated. What would be the way to compute this integral?