Consider the integral $\iint\left(2x^3y+xy^2\right)dA$ to be evaluated over a region limited by the equations $y=x^2+1$ , $y=x^2$, $xy=3$ and $xy=1$. 
I tried the fallowing substitutions: $y-x^2=v$ and $xy=u$. The limits of integration become simplified: $\left\{\left(u,v\right):\:1\le u\le 3,\:0\le v\le 1\right\}$. But solving for x and y and substituting back in the original funtion, I end up with a monstrous expression. Is there any other way?
You can perform the change $u=x\,y$ and $v=y-x^2$ with jacobian $du\,dv=(2x^2+y)\,dx\,dy$ and your integral reads
$$\int\int_D 2x^3y+x\,y^3\,dx\,dy=\int_1^3\int_0^1u\,dv\,du=4$$