Given
$$f(x,y) = \begin{cases} \frac{x^2 - y^2}{(x^2+y^2)^2}, &(x,y)\neq (0,0)\\ 0, &(x,y)=(0,0), \end{cases} $$
I need to calculate $\int_0^1\int_0^1 f(x,y)\, dx\,dy$ and $\int_0^1\int_0^1f(x,y)\,dy\,dx$. I know that the answers will be different, but I'm having trouble with the calculation part.
Even just advice on how to solve the single integral will be helpful. Thanks.
Consider
$$\int_0^1\left(\int_0^1\frac{x^2 - y^2}{(x^2 + y^2)^2}\, dy \right) \, dx \\ = \int_0^1\left(\int_0^1\frac{\partial}{\partial y}\left(\frac{y}{x^2 + y^2}\right)\, dy \right) \, dx \\ = \int_0^1 \frac{dx}{x^2+1}$$