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