A Short Tale of Long Tail Integration

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.