Past exam of the mathematics course of the University of Oxford / Undated / Prelims / Mathematics III resit / 2 / b

The Jacobian determinant isThe change of variables is defined only for $x\ne 0$. On the interior of the integration region, $x>0$ and $y>0$, so this Jacobian determinant is negative and nonzero.

The inverse change of variables is $x=v/(1+u)$, $y=uv/(1+u)$. Thus the triangular region becomesThe upper limit depends on $u$: the new region is not a rectangle. The absolute inverse Jacobian determinant is $x^2/(x+y)$, so the change of variables formula cancels the rational factor in the integrand:Although the original integrand is undefined at the vertex $x=y=0$, its nonnegative improper integral exists. One can apply the change of variables formula on $x\ge \epsilon$ and let $\epsilon \downarrow 0$ by the monotone convergence theorem; the transformed integrand has the finite integral just computed.