improper integrals of rational functions

Many integrals over the whole real line, like the integral from minus infinity to infinity of dx / (x^2 + 1) or dx / (x^4 + 1), are perfectly finite but stubborn to evaluate by real antiderivatives, especially for high-degree denominators. The contour method computes them by closing the real line into a loop in the complex plane and harvesting residues.

The strategy: take a real rational function P(x) / Q(x) with Q having no real zeros and the degree of Q at least 2 more than the degree of P (so the integral converges). Form the contour that runs along the real axis from -R to R and then closes with a large semicircle of radius R in the upper half-plane. The residue theorem says the whole loop integral equals 2 pi i times the sum of residues of P(z)/Q(z) at the poles in the upper half-plane. Now let R go to infinity: the semicircle's contribution shrinks to zero (because P/Q decays like 1/R^2 or faster while the arc length grows only like R, so the product vanishes — this is the degree-at-least-2 condition doing its job). What survives is exactly the real integral you wanted.

So the rule comes out clean: the integral from minus infinity to infinity of P(x)/Q(x) dx = 2 pi i times the sum of the residues of P(z)/Q(z) at the poles in the upper half-plane. You could close downward instead, which encircles the lower poles clockwise and introduces a minus sign; the answer is the same. This is the cleanest entry point to contour evaluation of real integrals, and the template generalizes to oscillatory integrands once you add Jordan's lemma.