Interpolando cinco puntos $(x_{0}, f(x_{0}))$, $(x_{0}+h, f(x_{0}+h))$, $(x_{0}+2h, f(x_{0}+2h))$, $(x_{0}+3h, f(x_{0}+3h))$ y $(x_{0}+4h, f(x_{0}+4h))$, mediante un polinomio de Lagrange e integrando
\begin{equation*} I = \frac{2}{45} h \ [7 f(x_{0}) + 32 f(x_{0}+h) + 12 f(x_{0}+2h) + 32 f(x_{0}+3h) + 7 f(x_{0}+4h)] \end{equation*}Usando la notación acostumbrada
\begin{equation*} I = \frac{2}{45} h \ [7 f(x_{i}) + 32 f(x_{i+1}) + 12 f(x_{i+2}) + 32 f(x_{i+3}) + 7 f(x_{i+4})] \end{equation*}
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