The boundary integral equation method or simply the Boundary Element Method (BEM) is now considered to be a powerful tool for solving problems in mechanics. A large number of the line and area integrals appearing for two-dimensional problems, in the BEM, can be represented as linear combinations of four singular functions. These integrals that are generated are products of the approximating polynomials and one or more of the four singular functions. These integrals can be evaluated numerically or analytically. The advantage of numerical integration is that the shape of the boundary can be of any complexity. The big disadvantage is that another source of error, besides the polynomial interpolation, is added into the process due to the approximation of the integrand for numerical integration. The singular nature of the fundamental solutions further exacerbates this disadvantage. In analytical integration, the form of the boundary must be assumed. The usual representation is a sum of straight-line segments. Currently, analytical expressions are available only for a few formulations and they are valid only for zero- and first-order polynomials. In this paper analytical expressions of the integrals are obtained for approximating polynomials of any arbitrary order. The fundamental solutions of the Laplace, Biharmonic and the equation of Plane Elastostatics are considered. The validity and advantage of using these integrals are shown by some simple problems in mechanics that have theoretical solutions.