Past exam of the mathematics course of the University of Oxford / 2021 / Prelims / Mathematics V / 1 / a / Solution

Use the change of variables $u=x^2-y^2$, $v=xy$. Its Jacobian determinant in the first quadrant is $2(x^2+y^2)>0$, and the inverse is uniquely determined byBoth are positive for $v>0$, so the transformation is one-to-one on the region. It becomes the rectangle $-2\le u\le 4$, $1\le v\le 3$. The integrand is $uv$, and $x^2+y^2=\sqrt{u^2+4v^2}$. Thus