Newton-Cotes closed quadrature ($w_j={\int }_0^s{L}_j(t)dt$)

Integrate the Lagrange polynomial through values at all equally spaced nodes $0,\dots ,s$, including both endpoints. The resulting quadrature rule is exact through degree $s$. For even $s$, the node polynomial ${\prod }_{j=0}^s(t-j)$ is odd about $s/2$, so its integral vanishes and exactness extends through degree $s+1$. The weights need not remain positive as the number of nodes increases.