Interpolando dos puntos $(x_{0}, f(x_{0}))$ y $(x_{0}+h, f(x_{0}+h))$, mediante un polinomio de Lagrange
\begin{equation*} f(x) = \frac{x - (x_{0}+h)}{x_{0} - (x_{0}+h)} f(x_{0}) + \frac{x - x_{0}}{(x_{0}+h) - x_{0}} f(x_{0}+h) \end{equation*}Integrando
\begin{equation*} \int_{x_{0}}^{x_{0}+h} \frac{x - (x_{0}+h)}{x_{0} - (x_{0}+h)} f(x_{0}) + \frac{x - x_{0}}{(x_{0}+h) - x_{0}} f(x_{0}+h) \ dx = \frac{1}{2} h \ [f(x_{0}) + f(x_{0}+h)] \end{equation*}Usando la notación acostumbrada
\begin{equation*} I = \frac{1}{2} h \ [f(x_{i}) + f(x_{i+1})] \end{equation*}
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