\int_{0}^{1}\int_{0}^{1}\frac{2xy}{x+y} \,dy\,dx = \int_{0}^{1}\int_{0}^{1} \frac {2xy + 2y^2 - 2y^2}{x+y} \,dy\,dx =
\int_{0}^{1}\int_{0}^{1}\frac{2y\left(x+y\right)-2y^2}{x+y} \,dy\,dx \\=
\int_{0}^{1}\int_{0}^{1} 2y- \frac{2y^2}{x+y}\,dy\,dx =
\int_{0}^{1}\int_{0}^{1}2y \,dy\,dx -2\int_{0}^{1}\int_{0}^{1}\frac{y^2}{x+y}\,dy\,dx=\dots =\frac{4}{3}\left(1-ln2\right)