I would like to solve the following double integral using the transformation:
$u=x+y$ , $v=x/y$ $$\int _0^1\int _y^2\frac{\left(x+y\right)}{x^2}\:e^{\left(x+y\right)}dxdy$$
What I've reached so far: $$J=-\frac{u}{\left(v+1\right)^2}$$ , $$x=\frac{uv}{v+1}$$, $$y=\frac{u}{v+1}$$ the only problem I'm facing now is to find the new region so I can set up my double integral, but what I get is: $$u=0,u=v+1,uv=2\left(v+1\right),v=1 $$ which doesn't look like a region that I can integrate over.
Final note: I know that the integral doesn't converge anyway, and I'm not the one who chose this substitution (homework problem)
If you graph the region bounded by your four curves you can divide it in two regions: $$ 1\leq v\leq 2,\ \ \ 0\leq u\leq v+1 $$ and $$ 2\leq v,\ \ \ 0\leq u\leq \frac{2(v+1)}v. $$