Past exam of the mathematics course of the University of Cambridge / 2018 / ib / Paper 1 / 13A / a

The integrandhas one pole inside the rectangle, at $z=0$, with residue $1$. The residue theorem therefore gives ${\int }_{C}F(z)dz=2\pi i$. The vertical-side integrals tend to zero as $R\to \infty$.

On the horizontal sides, direct substitution $z=x\pm \pi i$, with the upper side oppositely oriented, shows that their sum isTaking the limit and comparing with $2\pi i$ gives