$\newcommand{\d}{\mathrm{d}}$My friend give me an integral exercise, which says that:
Give area $D$ which is edged by line $x+y=1$ and $x$ axis and $y$ axis. Now calculate $$\iint_{D}\exp({\frac{y}{x+y}})\,\d x\,\d y.$$
I find that when $(x=0) \land (y=0)$, $\exp(\frac{y}{x+y})$ is not defined, and I don't know if it is integrable.
Additionally, I notice that $\iint_{D}\exp({\frac{y}{x+y}})\,\d x\,\d y = \iint_{D}\exp({\frac{x}{x+y}})\,\d x\,\d y$, and assume the result is $S$, we can know that \begin{align} S &= \frac{1}{2}\Bigg(\iint_{D}\exp({\frac{y}{x+y}})\,\d x\,\d y + \iint_{D}\exp({\frac{x}{x+y}})\,\d x \,\d y\Bigg) \\ &= \frac{1}{2}\iint_{D}\exp({\frac{y}{x+y}}) + \exp(\frac{x}{x+y})\,\d x\,\d y, \end{align} I try to use Green Theorem, and I try to calculate $Q=\int \exp(\frac{x}{x+y})\,\d x$ and $P=\int \exp(\frac{y}{x+y})\,\d x$, I try to construct the $\iint_{D} (\frac{\partial Q}{\partial x} -\frac{\partial P}{\partial y} )\,\d x\,\d y$. But I find that I can't calculate $P$ and $Q$.
Now, I have two question:
Is this exercise integrable?
How to calculate it? or how to modify my try on calculating?
ps：Thanks @StubburnAtom for your reminders. However, I find the answer of @Quanto in my question is different from all of other answers under those similar questions. So, if anyone find this question of mine, you can try to find all of other questions and find the answers there, and you can also look at the answer under this question, it is also a very good answer.
Rotate 45-degrees with $u= \frac1{\sqrt2}(x+y)$ and $v= \frac1{\sqrt2}(x-y)$ to transform the integral to $$\iint_{D}e^{\frac{y}{x+y}}dx dy = \int_0^{\frac1{\sqrt2}}\int_{-u}^u e^{\frac{u-v}{2u}}dv du =2(e-1) \int_0^{\frac1{\sqrt2}}u\>du=\frac12(e-1) $$