How should I solve this integral with changing parameters? I can't solve this. How should I proceed?


  $$\iint_De^{\large\frac{y-x}{y+x}}\mathrm dx\mathrm dy$$


$D$ is the triangle with these coordinates $(0,0), (0,2), (2,0)$ and I've changed the parameters this way $u=y-x$ and $v= y+x$ and the Jacobian is $-\frac{1}{2}$ but I have problem finding the range of $u$ and $v$ to calculate the integral.
Start by making a drawing of your domain. You can see that the $v$ is along the diagonal in the first quadrant, and $u$ is along the diagonal in the second quadrant. You can also see that the line between $(0,2)$ and $(2,0)$ is parallel to $u$, and intersects $v$ axis at $v=1$. So $v$ varies between $0$ and $1$ and $u$ varies between $-v$ and $v$.
Computer algebra gives (for the general case):

$$e \left(x^2 \text{Ei}\left(-\frac{2 x}{x+y}\right)-\frac{y^2 \text{Ei}\left(\frac{2
   y}{x+y}\right)}{e^2}\right)+\frac{1}{2} e^{1-\frac{2 x}{x+y}} (x+y)^2$$

over your specified region:

$$e-\frac{1}{e}$$