Abstract
In this paper, the Harmonic mean and Contra-harmonic mean of two endpoints [a,b] are applied at the derivative of the error term in the existing CNC formulas. This modified error term is included as an additional function to the existing formula. The new error terms are derived using the method of precision for the proposed formulas. Comparisons are made between the existing CNC formulas and the proposed CNC formulas by using numerical examples. The proposed Harmonic mean derivate-based (HMD) and the Contra-harmonic mean derivative-based (CHMD). CNC formulas give better results for definite integrals. Another two of its statistical means are discussed and derived for the existing CNC formulas in the subsequent paper.
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