Summary: It is difficult to evaluate definite integrals numerically when the integrand function is highly variable. The factor responsible for rapid variability can be substituted by a special analytical function and then a path of integration can be taken along any closed contour containing the interval \([-1,+1]\) in its interior. A contour avoiding all singularities of integrand may be chosen. An improper, Cauchy type singular integral of a rapidly changing integrand can be replaced by the integral of a continuous and smooth integrand. In some cases the finite formula can be obtained by applying Cauchy’s residue theorem.