Interpolando cinco puntos \( (x_{0}, f(x_{0})) \), \( (x_{1}, f(x_{1})) \), \( (x_{2}, f(x_{2})) \), \( (x_{3}, f(x_{3})) \) y \( (x_{4}, f(x_{4})) \), mediante un polinomio de Lagrange $$ \begin{equation*} f(x) = \begin{bmatrix} \cfrac{(x - x_{1})(x - x_{2})(x - x_{3})(x - x_{4})}{(x_{0} - x_{1})(x_{0} - x_{2})(x_{0} - x_{3})(x_{0} - x_{4})} \\ \cfrac{(x - x_{0})(x - x_{2})(x - x_{3})(x - x_{4})}{(x_{1} - x_{0})(x_{1} - x_{2})(x_{1} - x_{3})(x_{1} - x_{4})} \\ \cfrac{(x - x_{0})(x - x_{1})(x - x_{3})(x - x_{4})}{(x_{2} - x_{0})(x_{2} - x_{1})(x_{2} - x_{3})(x_{2} - x_{4})} \\ \cfrac{(x - x_{0})(x - x_{1})(x - x_{2})(x - x_{4})}{(x_{3} - x_{0})(x_{3} - x_{1})(x_{3} - x_{2})(x_{3} - x_{4})} \\ \cfrac{(x - x_{0})(x - x_{1})(x - x_{2})(x - x_{3})}{(x_{4} - x_{0})(x_{4} - x_{1})(x_{4} - x_{2})(x_{4} - x_{3})} \end{bmatrix}^{T} \begin{bmatrix} f(x_{0}) \\ f(x_{1}) \\ f(x_{2}) \\ f(x_{3}) \\ f(x_{4}) \end{bmatrix} \end{equation*} $$
Integrando $$ \begin{equation*} \int_{x_{0}}^{x_{4}} f(x) \ dx = \begin{bmatrix} -\frac{(x_{0} - x_{4}) \{ x_{0}^{3} - \frac{1}{4} x_{0}^{2} (5 x_{1} + 5 x_{2} + 5 x_{3} - 3 x_{4}) + x_{0} [\frac{1}{2} x_{4}^{2} - \frac{5}{6} x_{4} (x_{1} + x_{2} + x_{3}) + \frac{5}{3} x_{1} (x_{2} + x_{3}) + \frac{5}{3} x_{3} x_{2}] + \frac{1}{4} x_{4}^{3} - \frac{5}{12} x_{4}^{2} (x_{1} + x_{2} + x_{3}) + \frac{5}{6} x_{4} [x_{1} (x_{2} + x_{3}) + x_{2} x_{3}] - \frac{5}{2} x_{1} x_{2} x_{3} \}}{5 (x_{0} - x_{1})(x_{0} - x_{2})(x_{0} - x_{3})} \\ -\frac{(x_{0} - x_{4})^{3} [x_{0}^{2} - \frac{1}{3} x_{0} (5 x_{2} + 5 x_{3} - 4 x_{4}) + x_{4}^{2} - \frac{5}{3} x_{4} (x_{2} + x_{3}) + \frac{10}{3} x_{2} x_{3}]}{20 (x_{0} - x_{1})(x_{1} - x_{2})(x_{1} - x_{3})(x_{1} - x_{4})} \\ \frac{(x_{0} - x_{4})^{3} [x_{0}^{2} - \frac{1}{3} x_{0} (5 x_{1} + 5 x_{3} - 4 x_{4}) + x_{4}^{2} - \frac{5}{3} x_{4} (x_{1} + x_{3}) + \frac{10}{3} x_{1} x_{3}]}{20 (x_{0} - x_{2})(x_{1} - x_{2})(x_{2} - x_{3})(x_{2} - x_{4})} \\ -\frac{(x_{0} - x_{4})^{3} [x_{0}^{2} - \frac{1}{3} x_{0} (5 x_{1} + 5 x_{2} - 4 x_{4}) + x_{4}^{2} - \frac{5}{3} x_{4} (x_{1} + x_{2}) + \frac{10}{3} x_{1} x_{2}]}{20 (x_{0} - x_{3})(x_{1} - x_{3})(x_{2} - x_{3})(x_{3} - x_{4})} \end{bmatrix}^{T} \begin{bmatrix} f(x_{0}) \\ f(x_{1}) \\ f(x_{2}) \\ f(x_{3}) \\ f(x_{4}) \end{bmatrix} \end{equation*} $$