Integration method unknown $$
\int \frac{4}{(x+y)^{2}} \,\mathrm{d}y = \frac{-4}{x+y}
$$
How do you get to that answer? I've been looking over my formulas, tried to compute the power and integrate after but no success.

Make the substitution $u=x+y$. Since you are integrating with respect to $y$, $du=dy$ therefore you have 
$$\int \frac{4}{(y+x)^2}dy= \int \frac{4}{u^2}du=4\int \frac{du}{u^2}=\frac{-4}{u}=\frac{-4}{x+y}$$
