Calculate integral of function over a triangle

Question

Let $R$ be the triangle whose vertices are the points $(0,0), (0,1)$ and $(1,0)$. Calculate the following integral: \begin{align*} \iint_R e^{\frac{y-x}{y+x}} \,dx\,dy \end{align*}

I think that i have to do some change of variables but i can't figure it out which one should that be. Any help?

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Edit

I thought about changing to variables as $u = y-x$, $v= y+x$, so the jacobian would be $-2$ and now i have to solve this other integral:

\begin{align*} -2\iint_D e^{\frac{u}{v}} \,du\,dv \end{align*}

Which i don't know how to solve either.

Answer 1

Try $u = y - x$ and $v = y+x$, since those are the terms you have. This makes the integrand easy. The Jacobian should be $\pm 2$.

Feel free to comment if this doesn't work and I'll try to give a full answer.