Arithmetic Mean Derivative-Based Quartet Midpoint Rule

A definite integral that is difficult to solve analytically can be calculated using the numerical integration methods. The midpoint rule is a prominent rule for approximating definite integrals. This article discusses a version of the quartet midpoint rule that includes the derivative of the arithmetic mean . The proposed rule increases precision over the previous rules. Furthermore, the error term is obtained by using the concept of precision between quadrature and exact values. Finally, the proposed rule is more effective than the present rule, according to numerical simulation results.