An exact expression is derived for the finite-part integral fp\(\int_ sr^{-3}fdS\) over a triangular domain S, where r denotes the distance of the points of the triangle from one of its vertices and f is a linear function of the Cartesian coordinates. The more general case where r denotes the distance of the points of the triangle from another constant point inside (or outside) the triangle is also studied and numerical results are presented. These integrals appear during the numerical solution of plane crack problems (under constant or variable pressure distribution) in three-dimensional elasticity. The availability of exact expressions for these integrals will increase the accuracy of the numerical results and, simultaneously, it will permit us not to use cubature formulae over triangular domains for the numerical solution of the previous class of problems.