A note on certain generalizations of the midpoint rule (Q1318391)

For integrals with a weight function, \textit{D. Jagerman} [Math. Comput. 20, 79-89 (1966; Zbl 0137.337)] and \textit{F. Stetter} [ibid. 22, 661-663 (1968; Zbl 0162.477)] have each introduced generalizations of the midpoint rule, that preserve the equiweight property through weight- dependent node selection. The proof of convergence for bounded Riemann- integrable functions is completed by a consideration of the Stetter case. From an asymptotic error estimate, a corrected Jagerman rule is proposed. Error bounds in terms of derivatives are given for the Jagerman rules. The performance of the rules is illustrated numerically.