Some New Time and Cost-Efficient Quadrature Formulas to Compute Integrals Using Derivatives with Error Analysis

Computational mathematics relies heavily on numerical integration, which has many scientific and engineering-related applications. For functions with complicated derivatives in particular, traditional quadrature formulae, while effective, may need substantial processing resources. An innovative method for efficient quadrature formulae that use derivatives for better accuracy and less computing work is presented in this study. Rigid derivation guarantees mathematical correctness and practical usability of the presented approaches. To test how well these formulae hold up under different circumstances, we run them through a thorough error analysis. Research shows that new approaches are more efficient and accurate than traditional ones, making them a good fit for high-precision real-world applications. We go over some of the possible uses and constraints of these approaches, as well as some suggestions for where the field may go from here in terms of improving their use in various computational contexts.