On numerical improvement of open Newton–Cotes quadrature rules

In this paper, we discuss about numerical improvement of the Open Newton–Cotes integration rules that are in forms of:∫a=x-1b=xn+1=x-1+(n+2)hf(x)dx≃∑k=0nBk(n)f(x-1+(k+1)h).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 the integral 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 the above integration formula up to degree n + 2. In this way, some numerical tests are given to show the numerical superiority of our idea with respect to the usual Open Newton–Cotes integration rules.