For computing integral F(x) dx integrated from -1 to +1, employ (1/N) * SUM_j F(x_j) summed over the N points x_j = [j-(N-1)/2] * 2/sqrt(N^2-1) for j=0,1,2,...,N-1. This is equal weight, equal spacing, and exact if F(x) is a polynomial of degree<=3. The two extreme points are located at +-sqrt((N-1)/(N+1)). If you try to integrate x^4 with this method, you get additive error = (8/15) / (N^2-1).