I want to solve the integral
$\int\nolimits_0^1 \int\nolimits_0^{1-x} \exp \left(\frac{y}{x+y} \right) \; \mathrm dy \; \mathrm dx$
and I am given the hint to use the substitution $x+y=u$ and $y = uv$. Now, I've never done such multiple substitutions before and I figured I'd just try and if I'm not mistaken, after substitution the integral looks as follows:
$\iint u \cdot e^v \; \mathrm dv \; \mathrm du.$
Now - given I am correct - how do I have to adjust the integral bounds? For $v$ I set $uv = 1-x$ and after some substitutions, I ended up with $v=\frac{1-u}{2u}$. However, when I try to find an integral bound for $u$, I fail, because $u = \frac{1}{1-v}$ and I don't know how to substitute $v$ such that the bound becomes a number (it should be one, or shouldn't it?).
So my question is: What are the correct integral bounds and how do I find them? Also, is my integral without bounds correct? Thanks for any answers in advance.