Evaluate $\int_0^1dx \int_0^1\frac{x^2-y^2}{(x^2+y^2)^2}dy$ Let $f(x,y)=\frac{x^2-y^2}{(x^2+y^2)^2}$. How to evaluate$$\int_0^1dx \int_0^1f(x,y)dy$$ and$$\int_0^1dy \int_0^1f(x,y)dx$$
Just do the integrals in order. For example, 
$$\int_0^1 f(x, y) d x = \frac{-1}{1+y^2},$$ and integrating the RHS with respect to $y$ should be easy peasy.

The other integral is almost identical.
Let's take the first case
$$ \int_0^1dx $$ evaluated to $1$
and
 $$ \int_0^1f(x,y)dy $$ evaluated at $x$ as constant.

And to get the result, multiply the two above.