I'm trying to integrate $$\iint e^{\frac{x}{x+y}}\,dx\,dy$$ where $y \leq (1-x)$ and $0 \leq x,y \leq 1$.
I tried to define new variables as $u=x$ and $v=x+y$, but I can't solve this either. I have notice $\int e^{\frac{1}{x}}dx$ is not elementary function. so it really confuses me..
Your intuition is correct and will form the starting point of our computations. Your integral is $\int \limits _0 ^1 \Bbb d x \int \limits _0 ^{1-x} \Bbb d y \space \Bbb e ^{\frac x {x+y}}$. Make the change of variable $v = x+y$ in the inner integral, obtaining $\int \limits _0 ^1 \Bbb d x \int \limits _x ^1 \Bbb d v \space \Bbb e ^{\frac x v}$ and thus decoupling the denominator from the numerator (previously they were coupled by having a $x$ in common). As you have noted, it is impossible to compute the inner integral, therefore change the order of integration: $\int \limits _0 ^1 \Bbb d v \int \limits _0 ^v \Bbb d x \space \Bbb e ^{\frac x v} = \int \limits _0 ^1 \Bbb d v \space v (\Bbb e - 1) = \frac {\Bbb e -1} 2$.