The residue theorem implies that you can compute some integrals on the real line by closing up the contour in the complex plane and accounting for any poles that you've enclosed. The trick is to choose your contour so that the contribution to the integral of the new piece goes to zero as the contour gets larger and larger (think of a real piece which goes from -R to R, and a semicircle in the upper half-plane connecting those two points; as R->infty, the real part of the integral goes from -infty to infty). One reason the new piece may go to zero is that oscillating functions on the real line turn in to exponential decay as you go up or down along the imaginary axis (as you point out), so as the new piece of the contour moves up or down, its contribution to the integral gets exponentially smaller.

(This is hard to explain without pictures and formulas, but you can find some examples here: http://web.williams.edu/Mathematics/sjmiller/public_html/302...).