The second kind Chebyshev quadrature rules of 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 shown 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 the step size. There are various cases for the weight function w(x) that one can use. Here we consider the weight function w(x)=1-x2, which is important in Numerical Analysis. Therefore, in this paper, we would face with the following formula:∫-1+1f(x)1-x2dx≃∑k=0nwkf-1+2k(n+1).It is known that the precision degree of above formula is n + 1 for even n's and is n for odd n's. But we consider the upper and lower bounds as two additional variables to reach a nonlinear system that numerically improves the precision degree of above formula up to degree n + 2. In this way, we also give some examples to show the numerical superiority of our idea.