The problem of evaluating integrals having a singularity of the form \((t- x)^{-m}\), where \(m\geq 1\), is studied. Integrals with strong singularities are interpreted in the Hadamard sense and are used for obtaining approximate solutions of singular integral equations. A mixed boundary value problem from the theory of elasticity is considered as an example. Particularly for integral equations where the kernel contains, in addition to the dominant term \((t-x)^{-m}\), terms which become unbounded at the end points, the present technique appears to be extremely effective to obtain rapidly converging numerical results.