Past exam of the algebra, analysis and differential equations 1 course of the University of Oxford / 2006 / 5 / Solution

Apply the residue theorem to $F(z)=(z^2+4{)}^{-2}$ on the upper semicircle of radius $R>2$, traversed counterclockwise. Its sole enclosed pole is the double pole at $2i$, andOn the semicircular arc, $|z^2+4|\ge R^2-4$, so the ML inequality bounds its contribution by $\pi R/(R^2-4{)}^2\to 0$. The real integral is absolutely convergent, since its integrand is $O(|x{|}^{-4})$ at infinity. Taking $R\to \infty$ in the contour identity yields