The main issue of the extended finite element method (XFEM) is the numerical integration of the system of equilibrium equations. Indeed, in order to have a correct displacement jump vector, the integration needs to be achieved on both sides of the discontinuity and thus requires the existence of integration points on both sides of the discontinuity. A volume averaging based integration method is developed in the present work alleviating this constraint and applied to XFEM coupled with cohesive zone model in a three-dimensional formulation. Moreover, unlike other widely used integration methods, the proposed method does not require the a priori knowledge of the position of the discontinuity inside the finite element nor the projection of the state variables.