On numerical improvement of closed Newton-Cotes quadrature rules

This paper discusses on numerical improvement of the Newton-Cotes integration rules, which are in forms of:∫ab=a+nhf(x)dx≃∑k=0nBk(n)f(a+kh).It is known that the precision degree of above formula is n + 1 for even n's and is n for odd n's. However, if its bounds are considered as two additional variables (i.e. a and h in fact) we reach a nonlinear system that numerically improves the precision degree of above integration formula up to degree n + 2. In this way, some numerical examples are given to show the numerical superiority of our approach with respect to usual Newton-Cotes integration formulas.