This paper is devoted to the solution of plane problems of the theory of elasticity by the method of discontinuous displacement using finite-part integrals (FPI). Two different integral equations (a real one and a complex one) with FPI's are obtained for the plane of a body with cracks. This opens the way for using arbitrary approximations of displacement discontinuities. The article contains integral formulae for FPI's used in the approximation of displacement discontinuities by polynomials of any order for internal elements and by special functions accounting for the asymptotic behaviour for the boundary elements. Therefore, prerequisites for increasing the accuracy of computations are created. The results of numerical experiments carried out indicate that there is a sharp increase (by two orders of magnitude) in the accuracy of the solution of the crack problem in which the integral formulae in question are used.