Variational integrators are a special kind of numerical difference schemes for mechanical systems that are derived from discrete variational principles. They exhibit obvious superiority to classical numerical algorithms. As obviously shown in the discrete Euler-Lagrange equations， the general construction of variational integrators finally comes down to the calculation of partial derivatives of the discrete Lagrangian， which is an approximation of the action integral of the Lagrangian over a short time interval. Inspired by this fact， analytic calculation of these partial derivatives is carried out， which induces a new integral depending on the Euler-Lagrange equations. Therefore， the evaluation of the integral of the Lagrangian by using any quadrature rule can be transformed into the evaluation of the newly induced integral based on a series of interval nodes. If the local trajectory is further fitted by requiring that the Euler-Lagrange equations hold at these interval nodes， the new integral will vanish when being computed through any numerical method. And accordingly， the calculation of the partial derivative of the discrete Lagrangian is simplified to calculate the partial derivative of the Lagrangian with respect to the velocity. The resulting algorithms of the new approach， which combine both the continuous and discrete Euler-Lagrange equations， not only preserve those unique benefits of variational integrators， but also produce smaller local errors with similar accuracy being attained.