Answer
$$-\ln 2$$
Work Step by Step
Consider $u=x-y \implies x=\dfrac{u+v}{2}$ and $y=x+y\implies y=\dfrac{v-u}{2}$
and $Jacobin =|\dfrac{1}{2}|$
Therefore, $$\dfrac{1}{2} \iint_{D} uv^{-1} dA =\iint_{R}\dfrac{x-y}{x+y} dA\\=\dfrac{1}{2} \int_{-2}^{0} \int_2^4 uv^{-1} dv du \\=\dfrac{1}{2} \times [\dfrac{u^2}{2}]_{-2}^0 \times [\ln v]_{2}^4 \\=-\ln 2$$
