Usando \( f(x) = 1 \) $$ \begin{equation*} a_{0} f(x_{0}) + a_{1} f(x_{1}) + a_{2} f(x_{2}) + a_{3} f(x_{3}) = \int_{x_{0}}^{x_{3}} 1 \ dx \end{equation*} $$
Reemplazando valores e integrando $$ \begin{equation*} a_{0} + a_{1} + a_{2} + a_{3} = x_{3} - x_{0} \end{equation*} $$
Usando \( f(x) = x \) $$ \begin{equation*} a_{0} f(x_{0}) + a_{1} f(x_{1}) + a_{2} f(x_{2}) + a_{3} f(x_{3}) = \int_{x_{0}}^{x_{3}} x \ dx \end{equation*} $$
Reemplazando valores e integrando $$ \begin{equation*} a_{0} x_{0} + a_{1} x_{1} + a_{2} x_{2} + a_{3} x_{3} = \frac{1}{2} x_{3}^{2} - \frac{1}{2} x_{0}^{2} \end{equation*} $$
Usando \( f(x) = x^{2} \) $$ \begin{equation*} a_{0} f(x_{0}) + a_{1} f(x_{1}) + a_{2} f(x_{2}) + a_{3} f(x_{3}) = \int_{x_{0}}^{x_{3}} x^{2} \ dx \end{equation*} $$
Reemplazando valores e integrando $$ \begin{equation*} a_{0} x_{0}^{2} + a_{1} x_{1}^{2} + a_{2} x_{2}^{2} + a_{3} x_{3}^{2} = \frac{1}{3} x_{3}^{3} - \frac{1}{3} x_{0}^{3} \end{equation*} $$
Usando \( f(x) = x^{3} \) $$ \begin{equation*} a_{0} f(x_{0}) + a_{1} f(x_{1}) + a_{2} f(x_{2}) + a_{3} f(x_{3}) = \int_{x_{0}}^{x_{3}} x^{3} \ dx \end{equation*} $$
Reemplazando valores e integrando $$ \begin{equation*} a_{0} x_{0}^{3} + a_{1} x_{1}^{3} + a_{2} x_{2}^{3} + a_{3} x_{3}^{3} = \frac{1}{4} x_{3}^{4} - \frac{1}{4} x_{0}^{4} \end{equation*} $$
Formando un sistema de ecuaciones $$ \begin{equation*} \begin{bmatrix} 1 & 1 & 1 & 1 \\ x_{0} & x_{1} & x_{2} & x_{3} \\ x_{0}^{2} & x_{1}^{2} & x_{2}^{2} & x_{3}^{2} \\ x_{0}^{3} & x_{1}^{3} & x_{2}^{3} & x_{3}^{3} \end{bmatrix} \begin{bmatrix} a_{0} \\ a_{1} \\ a_{2} \\ a_{3} \end{bmatrix} = \begin{bmatrix} x_{3} - x_{0} \\ \frac{1}{2} x_{3}^{2} - \frac{1}{2} x_{0}^{2} \\ \frac{1}{3} x_{3}^{3} - \frac{1}{3} x_{0}^{3} \\ \frac{1}{4} x_{3}^{4} - \frac{1}{4} x_{0}^{4} \end{bmatrix} \end{equation*} $$
Resolviendo $$ \begin{align*} a_{0} &= -\frac{3 x_{0}^{3} - 4 x_{0}^{2} x_{1} - 4 x_{0}^{2} x_{2} - x_{0}^{2} x_{3} + 6 x_{0} x_{1} x_{2} + 2 x_{0} x_{1} x_{3} + 2 x_{0} x_{2} x_{3} - x_{0} x_{3}^{2} - 6 x_{1} x_{2} x_{3} + 2 x_{1} x_{3}^{2} + 2 x_{2} x_{3}^{2} - x_{3}^{3}}{12 (x_{2} - x_{0})(x_{1} - x_{0})} \\ a_{1} &= \frac{x_{0}^{4} - 2 x_{0}^{3} x_{2} - 2 x_{0}^{3} x_{3} + 6 x_{0}^{2} x_{2} x_{3} - 6 x_{0} x_{2} x_{3}^{2} + 2 x_{0} x_{3}^{3} + 2 x_{2} x_{3}^{3} - x_{3}^{4}}{12(x_{1} - x_{0})(x_{3} - x_{1})(x_{2} - x_{1})} \\ a_{2} &= \frac{x_{0}^{4} - 2 x_{0}^{3} x_{1} - 2 x_{0}^{3} x_{3} + 6 x_{0}^{2} x_{1} x_{3} - 6 x_{0} x_{1} x_{3}^{2} + 2 x_{0} x_{3}^{3} + 2 x_{1} x_{3}^{3} - x_{3}^{4}}{12(x_{0} x_{1} - x_{0} x_{2} - x_{1} x_{2} + x_{2}^{2})(x_{2} - x_{3})} \\ a_{3} &= -\frac{x_{0}^{3}- 2 x_{0}^{2} x_{1} - 2 x_{0}^{2} x_{2} + x_{0}^{2} x_{3} + 6 x_{0} x_{1} x_{2} - 2 x_{0} x_{1} x_{3} - 2 x_{0} x_{2} x_{3} + x_{0} x_{3}^{2} - 6 x_{1} x_{2} x_{3} + 4 x_{1} x_{3}^{2} + 4 x_{2} x_{3}^{2} - 3 x_{3}^{3}}{12(x_{1} x_{2} - x_{1} x_{3} - x_{2} x_{3} + x_{3}^{2})} \end{align*} $$