Some integral inequalities for functions with (n−1)st derivatives of bounded variation

In this paper, we generalize Cerone’s results, and a unified treatment of error estimates for a general inequality satisfying f(n−1)f(n−1) being of bounded variation is presented. We derive the estimates for the remainder terms of the mid-point, trapezoid, and Simpson formulas. All constants of the errors are sharp. Applications in numerical integration are also given.