A numerical quadrature algorithm is developed, for integrands which may exhibit some kind of singular behaviour within the finite of infinite integration range. Using the automatical FORTRAN IV integration program, one should provide the abscissae the function is not “smooth” at. The quadrature formula has been obtained by applying the trapezoidal rule after transformation of the integrand. Standing severe tests which were based on the test functions of Casaletto et al. and on Kahaner's sample set, the integration scheme turned out to be of a remarkable reliability, efficiency and accuracy.