A singular integral equation (with a Cauchy type kernel) can be solved directly by the quadrature method, which consists in approximating the integral terms by an appropriate interpolatory or Gaussian quadrature rule and, next, using appropriately selected collocation points for the reduction of the singular integral equation to a system of linear algebraic equations. In this paper, it is proposed that the simple poles of the right-hand side function of the singular integral equation (if they exist) be taken into account during the numerical integrations. The cases of the Gauss- and Lobatto-Chebyshev methods are considered in detail and two numerical applications (concerning crack problems) are presented. These applications show the significant improvement of the numerical results (under appropriate circumstances) if the present results are taken into consideration. Finally, further generalizations of these results are reported in brief.