The first kind Chebyshev-Newton-Cotes quadrature rules (closed type) and its numerical improvement

One of the less-known integration methods is the weighted Newton-Cotes of closed type quadrature rule, which is denoted by:∫a=x0b=xn=x0+nhf(x)w(x)dx≃∑k=0nwkf(x0+kh),where w(x) is a positive function and h=b-an is a positive value. There are various cases for the weight function w(x) that one can use. Because of special importance of the weight function of Gauss-Chebyshev quadrature rules, i.e. w(x)=11-x2 in numerical analysis, we consider this function as the main weight. Hence, in this paper, we face with the following formula in fact:∫-1+1f(x)1-x2dx≃∑k=0nwkf-1+2kn.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 we consider its bounds as two additional variables we reach a nonlinear system that numerically improves the precision degree of above formula up to degree n + 2. In this way, we give several examples which show the numerical superiority of our approach.