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

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