  Integration of the form $\int_a^\infty {f(x)w(x)dx} $, where $w(x)$ is either
$\sin (\omega {\kern 1pt} x)$ or $\cos (\omega {\kern 1pt} x)$, is widely
encountered in many engineering and scientific applications, such as those
involving Fourier or Laplace transforms. Often such integrals are approximated
by a numerical integration over a finite domain $(a,\,b)$, leaving a truncation
error equal to the tail integration $\int_b^\infty {f(x)w(x)dx} $ in addition
to the discretization error. This paper describes a very simple, perhaps the
simplest, end-point correction to approximate the tail integration, which
significantly reduces the truncation error and thus increases the overall
accuracy of the numerical integration, with virtually no extra computational
effort. Higher order correction terms and error estimates for the end-point
correction formula are also derived. The effectiveness of this one-point
correction formula is demonstrated through several examples.
