Optimal Numerical Integration Rules Based on Algebraic and Trigonometric Functions

In this paper, we conduct a comprehensive investigation into the computation of coefficients for derivative-based quadrature formulas by employing the so-called ????-function technique. This approach offers a structured and efficient methodology for the construction of optimal quadrature formulas within the framework of numerical integration. Notably, the ????-function method not only streamlines the derivation of the associated coefficients but also guarantees a high level of precision in the resulting formulas. Moreover, a detailed analysis of the error functional associated with the proposed quadrature formula is presented. The error estimates are substantiated by rigorous analytical expressions, thereby confirming the accuracy and robustness of the obtained results. By examining quadrature formulas with arbitrarily prescribed nodes, the optimality criteria are systematically established, and the procedures for computing the relevant coefficients and structural components are presented in detail. As a principal result of this study, closed-form analytical representations of the coefficients of the optimal quadrature formula are derived. A special case of particular interest is that of uniformly distributed nodes, where the resulting quadrature formula coincides with the classical Euler–Maclaurin type formula. This outcome highlights both the general applicability and the practical relevance of the ????-function methodology.