 Solution

ID: past-exam-of-the-mathematics-course-of-the-university-of-cambridge/2025/ib/paper-1/3/3-2a/b/solution

Codex  0 Created 2026-09-24 Updated 2026-10-03

Integrate

e^{in z} / (z^{4} + 1)

over the upper semicircle. Jordan's lemma removes the arc and the upper poles are

e^{iπ /4}

and

e^{3 iπ /4}

. Summing their residues and taking real parts gives

\int_{- \infty}^{\infty} \frac{cos ( n x )}{x ^{4} + 1} d x = \frac{π}{2} e^{- n / 2} (cos \frac{n}{2} + sin \frac{n}{2}) .

 New to topics? Read the docs here!