We take an example of applying the Cauchy residue theorem in evaluating usual real improper integrals.

We shall calculate the integral

where $k$ is any real number.  One may prove that the integrand has no antiderivative among the elementary functions if  $k\neq 0$.

Since the integrand is an even and  $x\mapsto\frac{\sin{kx}}{1+x^{2}}$  an odd function, we may write

using also Euler’s formula.  Let’s consider the contour integral

where $\gamma$ is the perimeter of the semicircle consisting of the line segment from $(-R,\,0)$ to $(R,\,0)$ and the semi-circular arc $c$ connecting these points in the upper half-plane ($R>1$).  The integrand is analytic on and inside of $\gamma$ except in the point  $z=i$  which is a simple pole.  Because we have (cf. the coefficients of Laurent series)

the residue theorem yields

This does not depend on the radius $R$ of the circle.

We split the integral $J$ in two portions:  one along the diameter and the other along the circular arc $c$.  So we obtain

When  $R\to\infty$,  the former portion tends to the limit $2I$ and the latter — as we at once shall see — to the limit 0.  Hence we get the result

As for the latter part of $J$, we denote  $z:=x+iy$ ($x,\,y\in\mathbb{R}$);  then on the arc $c$, where  $|z|=R$  and  $y\geqq 0$, we have

Using this estimation of the integrand we get, according the integral estimating theorem, the inequality

Since the right hand member tends to 0 as  $R\to\infty$, then also the left hand member.