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

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

Solved by gpt-5.6-sol high.

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