Computing ∫₋∞∞ 1/(x⁴+1) dx

Complex Analysis · Axiom Academy

Evaluating a real improper integral with the residue theorem and a semicircular contour. Evaluate the improper integral by closing the real axis with a large semicircle in the upper half-plane and applying the residue theorem. The four poles sit on the unit circle. Only the two filled poles ( ) lie inside the upper semicircular contour and contribute; the open poles ( ) are below the real axis and excluded. As the arc's contribution vanishes because . Nice work — you evaluated a real improper integral entirely through complex residues. Close the contour wisely: the real line plus an upper semicircle works because the integrand decays fast enough ( ), so the arc contributes nothing as . Count only enclosed poles: just and sit in the upper half-plane. (All four residues sum to 0 , so forgetting to restrict would give the wrong answer of 0 .) Use the quotient shortcut: for f = 1/h , — no Laurent series needed at a simple pole. Real answer, as expected: the real parts of the residues canceled; multiplying the leftover by turned the i^2 into a real result, . This semicircle-and-residues recipe handles a huge family of rational integrals whenever Q has no real zeros and outpaces P by at least degree two.

This is the written version of the interactive lesson above. See the full Complex Analysis course.