We have derived stress integral equations for an arbitrary shaped plane crack in a three-dimensional (3D) infinite homogeneous isotropic elastic medium. First we obtained the integral equation for static crack, which is consistent with previous works. Then we derived a new boundary integral equation for the study of dynamic tensile and shear cracks. We removed the hypersingularity that appears in the usual formulation of the stress-boundary integral equation method by a regularization technique. The dynamic stress integral equation consists of three terms which can be physically explained. The first term is a modification of the static solution, the second one represents the diffraction of waves at slip discontinuities, and the last term is related to the effect of an instantaneous change in slip velocity. The first term controls the overall shape of the stress drop inside the crack. This term is important when we consider the final stress drop on the fault. However, the second and the third terms are dominant at a moving crack tip, and are responsible for the radiation of P and S waves. They are also very important for the application of a fracture criterion at the rupture front. Using this boundary integral equation we have calculated several numerical examples for both the static and dynamic fault models.