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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)

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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 $$ 2\leq v,\ \ \ 0\leq u\leq \frac{2(v+1)}v. $$

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  • $\begingroup$ I'm afraid that the sign in the straight line $u=v-1$ should be a plus, tried editing but that's just one character. $\endgroup$ – Malzahar Dec 14 '19 at 19:32
  • $\begingroup$ Done, thanks. $ $ $\endgroup$ – Martin Argerami Dec 14 '19 at 21:28

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