Past exam of the mathematics course of the University of Cambridge / 2025 / ib / Paper 1 / 3 / 3.2A / b / Solution

Integrate $e^{inz}/(z^4+1)$ over the upper semicircle. Jordan's lemma removes the arc and the upper poles are $e^{i\pi /4}$ and $e^{3i\pi /4}$. Summing their residues and taking real parts gives