Use contour integration to evaluate the integral over the whole real line of dx/(1 + x^4). Give the poles you enclose, their residues, and the final value.

Use contour integration to evaluate the integral over the whole real line of dx/(1 + x^4). Give the poles you enclose, their residues, and the final value.

Approach: Close the contour with a large semicircle in the upper half plane, show the arc contribution vanishes, and sum the residues at the two fourth roots of minus one that lie inside.

pi/sqrt(2). The poles are the fourth roots of -1, and the two in the upper half plane are z_1 = e^{i pi/4} and z_2 = e^{3 i pi/4}. At a simple pole of 1/(1+z^4) the residue is 1/(4 z^3) = z/(4 z^4) = -z/4, using z^4 = -1, so the residues are -z_1/4 and -z_2/4. Their sum is -(e^{i pi/4} + e^{3 i pi/4})/4 = -(i sqrt(2))/4, since the real parts cancel and the imaginary parts add. The residue theorem gives the integral as 2 pi i times that sum, which is 2 pi i * (-i sqrt(2)/4) = pi sqrt(2)/2 = pi/sqrt(2) = 2.2214. The semicircular arc of radius R contributes at most pi R / (R^4 - 1), which tends to zero, so the closed contour and the real line agree in the limit.

Follow-up: How does the same method handle the integral of dx/(1 + x^6), and what is the general answer for dx/(1 + x^{2n})?

Key concepts: residue theorem, contour integration, roots of unity, semicircular arc.