Interpolando cuatro puntos \( (x_{0}, f(x_{0})) \), \( (x_{0}+h, f(x_{0}+h)) \), \( (x_{0}+2h, f(x_{0}+2h)) \) y \( (x_{0}+3h, f(x_{0}+3h)) \), mediante un polinomio de Lagrange e integrando $$ \begin{equation*} I = \frac{3}{8} h \ [f(x_{0}) + 3 f(x_{0}+h) + 3 f(x_{0}+2h) + f(x_{0}+3h)] \end{equation*} $$
Usando la notación acostumbrada $$ \begin{equation*} I = \frac{3}{8} h \ [f(x_{i}) + 3 f(x_{i+1}) + 3 f(x_{i+2}) + f(x_{i+3})] \end{equation*} $$