Show that $\int^1_0 \int^{1-x}_0e^\frac{y}{x+y}dy\,dx=\frac{e-1}{2}$
This is probably very simple, but I was instructed to use the substitution $x+y=u, y=uv$.
So by manipulating the integral and calculating the Jacobian (which is simply $u$, ie. $dx\,dy=u\,du\,dv$, I can express the transformation as $$\int \int ue^v dv \,du$$
The problem is that I don't know what to plug in as the limits of integration. For example, the limit $1-x$ in the expression $\int^1_0 \int^{1-x}_0e^\frac{y}{x+y}dy\,dx$ cannot be expressed as either a function of v or u purely with the substitution given. Am I missing something here? Thanks for the help.