This paper deals with the numerical study of stability and instantaneous extension of cracks under Griffith criterion, within the classical linear fracture mechanics framework. First- and second-order domain derivatives of the potential energy W in crack extensions play a central role in the mathematical formulation of such situations. The proposed solution strategy (called the θ-integral method, by reference to the FEM-based θ-method) possesses the following main characteristics: (i) it is based on a symmetric Galerkin boundary integral formulations; and (ii) explicit expressions for the domain derivatives of W are established using Lagrangian formulas (so that the derivations do not increase crack front singularities). The numerical solution procedure for the extension velocity problem, including all domain derivative evaluations, is entirely supported by the boundary element mesh of the current crack configuration; numerical differentiation is avoided. Numerical examples for an isolated crack in a 3-D unbounded elastic body show that in practice an excellent accuracy can be achieved for the energy release rate, the second-order domain derivatives of W and the prediction of stability or instability of crack growth.