I want to find $\iint_Dxye^yd(x,y)$, where $D=\{(x,y)\in\mathbb{R}^2|1\le x\le3, 0\le y\le \ln(x)\}$. To solve this, $D$ should me mapped to a rectangle, such that you can use Fubini's theorem and thus, reduce it all to simple integration on a rectangle. However, I can't think of a transformation $g:\mathbb{R}^2\to\mathbb{R}^2$, such that for an $\Omega\subset\mathbb{R}^2$ $\iint_Dxye^yd(x,y)=\iint_{g(\Omega)}xye^yd(x,y)$
So how to find $g$ and $\Omega$?
Have you tried $$ (x, y) \mapsto (x, y \ln x) ? $$
This takes the rectangle $$ 1 \le x \le 3 \\ 0 \le y \le 1 $$ to $$ 1 \le x \le 3 \\ 0 \le y \le \ln x . $$