Optimization is a well-established tool for enhancing the performance of dynamical systems. Among the available methods, gradient-based optimization techniques are often the most efficient. However, their application requires sensitivity information that depends on the optimization problem. In the field of multibody dynamics, the discrete adjoint variable method provides an effective approach to perform the sensitivity analysis and has shown to be more efficient and more accurate than the continuous adjoint method. In the literature, time integration techniques based on the Newmark method are usually used to solve the primal problem and the integral criterion function is discretized by the trapezoidal rule. When applying the discrete adjoint method to this, the resulting gradient of the criterion function is typically second-order accurate. Yet, if higher-order integration methods are applied, the low order of the trapezoidal rule dominates the convergence of the gradient even though the solution of the primal problem is highly accurate. This effect is intensified by employing adaptive step size control as high order integration methods tend to perform large time steps over which the evaluation of the integral criterion becomes more difficult In this contribution, the application of different quadrature rules to the discrete adjoint sensitivity analysis for both the Newmark method and the Runge-Kutta method is investigated. A central aspect of this contribution is whether the accuracy and the convergence order of the discrete dual problem can be improved by applying higher-order quadrature rules which are evaluated using interpolations of the primal problem.