<s>[INST] How to integrate $(x^2 - y^2) / (x^2 + y^2)^2$ How do I integrate 
$$\int \int \frac{(x^2 - y^2)}{(x^2 + y^2)^2} dx dy?$$
The WolframAlpha page gives 
$$
c_1 + c_2 + \tan^{-1}(x/y).
$$
And I kind of specifically need
$$
\int_{0}^{x} \frac{(x^2 - y^2)}{(x^2 + y^2)^2} dy.
$$
Note


*

*I want to know integration technique to solve this without using $F' = f$.

*For the double integral above, what I'm interested is Lebesgue integral, but I guess what Wolfram gave is in the Riemann sense.

 [/INST] By the quotient rule $$\frac{\partial}{\partial y}\left(\frac{y}{x^2 + y^2}\right) = \frac{x^2 - y^2}{(x^2 + y^2)^2}.$$
Therefore 
$$\int_0^x \frac{x^2 - y^2}{(x^2 + y^2)^2}\, dy = \int_0^x \frac{\partial}{\partial y}\left(\frac{y}{x^2 + y^2}\right)\, dy = \frac{y}{x^2 + y^2}\bigg|_{y = 0}^{y = x} = \frac{1}{2x}.$$
</s>