Given the region $R=\{(x, y): x>0, y>0, x+y<a\}$, Evaluate $$I=\iint_{W}(x+y)^{-3 / 2} \exp \left(\frac{x-y}{x+y}\right) d x d y$$
I tried using change of variables. $x+y=u$ and $y=v$ Then we have $0 \lt u \lt a$ and since $x=u-y=u-v>0$ we have $0<v<u$
The Jacobian will be $$J=\begin{vmatrix} \frac{\partial x}{\partial u} &\frac{\partial x}{\partial v} \\ \frac{\partial y}{\partial u}& \frac{\partial y}{\partial v} \end{vmatrix}=\begin{vmatrix} 1 &-1 \\ 0&1 \end{vmatrix}=1$$ Thus our integral will be: $$I=\int_{u=0}^a\int_{v=0}^{u}u^{\frac{-3}{2}} \times e\times e^{\frac{-2v}{u}}dudv$$ Is this change of variables fine?