Usando \( f(x) = 1 \) $$ \begin{equation*} a_{0} f(x_{0}) + a_{1} f(x_{1}) + a_{2} f(x_{2}) + a_{3} f(x_{3}) + a_{4} f(x_{4}) = \int_{x_{0}}^{x_{4}} 1 \ dx \end{equation*} $$
Reemplazando valores e integrando $$ \begin{equation*} a_{0} + a_{1} + a_{2} + a_{3} + a_{4} = x_{4} - x_{0} \end{equation*} $$
Usando \( f(x) = x \) $$ \begin{equation*} a_{0} f(x_{0}) + a_{1} f(x_{1}) + a_{2} f(x_{2}) + a_{3} f(x_{3}) + a_{4} f(x_{4}) = \int_{x_{0}}^{x_{4}} x \ dx \end{equation*} $$
Reemplazando valores e integrando $$ \begin{equation*} a_{0} x_{0} + a_{1} x_{1} + a_{2} x_{2} + a_{3} x_{3} + a_{4} x_{4} = \frac{1}{2} x_{4}^{2} - \frac{1}{2} x_{0}^{2} \end{equation*} $$
Usando \( f(x) = x^{2} \) $$ \begin{equation*} a_{0} f(x_{0}) + a_{1} f(x_{1}) + a_{2} f(x_{2}) + a_{3} f(x_{3}) + a_{4} f(x_{4}) = \int_{x_{0}}^{x_{4}} x^{2} \ dx \end{equation*} $$
Reemplazando valores e integrando $$ \begin{equation*} a_{0} x_{0}^{2} + a_{1} x_{1}^{2} + a_{2} x_{2}^{2} + a_{3} x_{3}^{2} + a_{4} x_{4}^{2} = \frac{1}{3} x_{4}^{3} - \frac{1}{3} x_{0}^{3} \end{equation*} $$
Usando \( f(x) = x^{3} \) $$ \begin{equation*} a_{0} f(x_{0}) + a_{1} f(x_{1}) + a_{2} f(x_{2}) + a_{3} f(x_{3}) + a_{4} f(x_{4}) = \int_{x_{0}}^{x_{4}} x^{3} \ dx \end{equation*} $$
Reemplazando valores e integrando $$ \begin{equation*} a_{0} x_{0}^{3} + a_{1} x_{1}^{3} + a_{2} x_{2}^{3} + a_{3} x_{3}^{3} + a_{4} x_{4}^{3} = \frac{1}{4} x_{4}^{4} - \frac{1}{4} x_{0}^{4} \end{equation*} $$
Usando \( f(x) = x^{4} \) $$ \begin{equation*} a_{0} f(x_{0}) + a_{1} f(x_{1}) + a_{2} f(x_{2}) + a_{3} f(x_{3}) + a_{4} f(x_{4}) = \int_{x_{0}}^{x_{4}} x^{4} \ dx \end{equation*} $$
Reemplazando valores e integrando $$ \begin{equation*} a_{0} x_{0}^{4} + a_{1} x_{1}^{4} + a_{2} x_{2}^{4} + a_{3} x_{3}^{4} + a_{4} x_{4}^{4} = \frac{1}{5} x_{4}^{5} - \frac{1}{5} x_{0}^{5} \end{equation*} $$
Formando un sistema de ecuaciones $$ \begin{equation*} \begin{bmatrix} 1 & 1 & 1 & 1 & 1 \\ x_{0} & x_{1} & x_{2} & x_{3} & x_{4} \\ x_{0}^{2} & x_{1}^{2} & x_{2}^{2} & x_{3}^{2} & x_{4}^{2} \\ x_{0}^{3} & x_{1}^{3} & x_{2}^{3} & x_{3}^{3} & x_{4}^{3} \\ x_{0}^{4} & x_{1}^{4} & x_{2}^{4} & x_{3}^{4} & x_{4}^{4} \end{bmatrix} \begin{bmatrix} a_{0} \\ a_{1} \\ a_{2} \\ a_{3} \\ a_{4} \end{bmatrix} = \begin{bmatrix} x_{4} - x_{0} \\ \frac{1}{2} x_{4}^{2} - \frac{1}{2} x_{0}^{2} \\ \frac{1}{3} x_{4}^{3} - \frac{1}{3} x_{0}^{3} \\ \frac{1}{4} x_{4}^{4} - \frac{1}{4} x_{0}^{4} \\ \frac{1}{5} x_{4}^{5} - \frac{1}{5} x_{0}^{5} \end{bmatrix} \end{equation*} $$
Resolviendo $$ \begin{align*} \tiny a_{0} &= \tiny -\frac{12 x_{0}^{4} - 15 x_{0}^{3} x_{1} - 15 x_{0}^{3} x_{2} - 15 x_{0}^{3} x_{3} - 3 x_{0}^{3} x_{4} + 20 x_{0}^{2} x_{1} x_{2} + 20 x_{0}^{2} x_{1} x_{3} + 5 x_{0}^{2} x_{1} x_{4} + 20 x_{0}^{2} x_{2} x_{3} + 5 x_{0}^{2} x_{2} x_{4} + 5 x_{0}^{2} x_{3} x_{4} - 3 x_{0}^{2} x_{4}^{2} - 30 x_{0} x_{1} x_{2} x_{3} - 10 x_{0} x_{1} x_{2} x_{4} - 10 x_{0} x_{1} x_{3} x_{4} + 5 x_{0} x_{1} x_{4}^{2} - 10 x_{0} x_{2} x_{3} x_{4} + 5 x_{0} x_{2} x4^{2} + 5 x_{0} x_{3} x_{4}^{2} - 3 x_{0} x4^{3} + 30 x_{1} x_{2} x_{3} x_{4} - 10 x_{1} x_{2} x_{4}^{2} - 10 x_{1} x_{3} x_{4}^{2} + 5 x_{1} x_{4}^{3} - 10 x_{2} x_{3} x_{4}^{2} + 5 x_{2} x_{4}^{3} + 5 x_{3} x_{4}^{3} - 3 x_{4}^{4}}{60 (x_{0} - x_{3})(x_{0} - x_{2})(x_{0} - x_{1})} \\ \tiny a_{1} &= \tiny -\frac{3 x_{0}^{5} - 5 x_{0}^{4} x_{2} - 5 x_{0}^{4} x_{3} - 5 x_{0}^{4} x_{4} + 10 x_{0}^{3} x_{2} x_{3} + 10 x_{0}^{3} x_{2} x_{4} + 10 x_{0}^{3} x_{3} x_{4} - 30 x_{0}^{2} x_{2} x_{3} x_{4} + 30 x_{0} x_{2} x_{3} x_{4}^{2} - 10 x_{0} x_{2} x_{4}^{3} - 10 x_{0} x_{3} x_{4}^{3} + 5 x_{0} x_{4}^{4} - 10 x_{2} x_{3} x_{4}^{3} + 5 x_{2} x_{4}^{4} + 5 x_{3} x_{4}^{4} - 3 x_{4}^{5}}{60 (x_{0} - x_{1})(x_{1} - x_{4})(x_{1} - x_{3})(x_{1} - x_{2})} \\ \tiny a_{2} &= \tiny \frac{3 x_{0}^{5} - 5 x_{0}^{4} x_{1} - 5 x_{0}^{4} x_{3} - 5 x_{0}^{4} x_{4} + 10 x_{0}^{3} x_{1} x_{3} + 10 x_{0}^{3} x_{1} x_{4} + 10 x_{0}^{3} x_{3} x_{4} - 30 x_{0}^{2} x_{1} x_{3} x_{4} + 30 x_{0} x_{1} x_{3} x_{4}^{2} - 10 x_{0} x_{1} x_{4}^{3} - 10 x_{0} x_{3} x_{4}^{3} + 5 x_{0} x_{4}^{4} - 10 x_{1} x_{3} x_{4}^{3} + 5 x_{1} x_{4}^{4} + 5 x_{3} x_{4}^{4} - 3 x_{4}^{5}}{ 60(x_{0} x_{1} - x_{0} x_{2} - x_{1} x_{2} + x_{2}^{2})(x_{2} - x_{4})(x_{2} - x_{3})} \\ \tiny a_{3} &= \tiny -\frac{3 x_{0}^{5} - 5 x_{0}^{4} x1 - 5 x_{0}^{4} x_{2} - 5 x_{0}^{4} x_{4} + 10 x_{0}^{3} x_{1} x_{2} + 10 x_{0}^{3} x_{1} x_{4} + 10 x_{0}^{3} x_{2} x_{4} - 30 x_{0}^{2} x_{1} x_{2} x_{4} + 30 x_{0} x_{1} x_{2} x_{4}^{2} - 10 x_{0} x_{1} x_{4}^{3} - 10 x_{0} x_{2} x_{4}^{3} + 5 x_{0} x_{4}^{4} - 10 x_{1} x_{2} x_{4}^{3} + 5 x_{1} x_{4}^{4} + 5 x_{2} x_{4}^{4} - 3 x_{4}^{5}}{60 (x_{0} x_{1} x_{2} - x_{0} x_{1} x_{3} - x_{0} x_{2} x_{3} + x_{0} x_{3}^{2} - x_{1} x_{2} x_{3} + x_{1} x_{3}^{2} + x_{2} x_{3}^{2} - x_{3}^{3})(x_{3} - x_{4})} \\ \tiny a_{4} &= \tiny \frac{3 x_{0}^{4} - 5 x_{0}^{3} x_{1} - 5 x_{0}^{3} x_{2} - 5 x_{0}^{3} x_{3} + 3 x_{0}^{3} x_{4} + 10 x_{0}^{2} x_{1} x_{2} + 10 x_{0}^{2} x_{1} x_{3} - 5 x_{0}^{2} x_{1} x_{4} + 10 x_{0}^{2} x_{2} x_{3} - 5 x_{0}^{2} x_{2} x_{4} - 5 x_{0}^{2} x_{3} x_{4} + 3 x_{0}^{2} x_{4}^{2} - 30 x_{0} x_{1} x_{2} x_{3} + 10 x_{0} x_{1} x_{2} x_{4} + 10 x_{0} x_{1} x_{3} x_{4} - 5 x_{0} x_{1} x_{4}^{2} + 10 x_{0} x_{2} x_{3} x_{4} - 5 x_{0} x_{2} x_{4}^{2} - 5 x_{0} x_{3} x_{4}^{2} + 3 x_{0} x_{4}^{3} + 30 x_{1} x_{2} x_{3} x_{4} - 20 x_{1} x_{2} x_{4}^{2} - 20 x_{1} x_{3} x_{4}^{2} + 15 x_{1} x_{4}^{3} - 20 x_{2} x_{3} x_{4}^{2} + 15 x_{2} x_{4}^{3} + 15 x_{3} x_{4}^{3} - 12 x_{4}^{4}}{60 (x_{1} x_{2} x_{3} - x_{4} x_{1} x_{2} - x_{4} x_{1} x_{3} + x_{4}^{2} x_{1} - x_{4} x_{2} x_{3} + x_{4}^{2} x_{2} + x_{4}^{2} x_{3} - x_{4}^{3})} \end{align*} $$