Usando \( f(x) = 1 \) $$ \begin{equation*} a_{0} f(x_{0}) + a_{1} f(x_{0}+h) = \int_{x_{0}}^{x_{0}+h} 1 \ dx \end{equation*} $$
Reemplazando valores e integrando $$ \begin{equation*} a_{0} + a_{1} = h \end{equation*} $$
Usando \( f(x) = x \) $$ \begin{equation*} a_{0} f(x_{0}) + a_{1} f(x_{0}+h) = \int_{x_{0}}^{x_{0}+h} x \ dx \end{equation*} $$
Reemplazando valores e integrando $$ \begin{equation*} a_{0} x_{0} + a_{1} (x_{0}+h) = \frac{1}{2} (x_{0}+h)^{2} - \frac{1}{2} x_{0}^{2} \end{equation*} $$
Formando un sistema de ecuaciones $$ \begin{equation*} \begin{bmatrix} 1 & 1 \\ x_{0} & x_{0}+h \end{bmatrix} \begin{bmatrix} a_{0} \\ a_{1} \end{bmatrix} = \begin{bmatrix} h \\ \frac{1}{2} (x_{0}+h)^{2} - \frac{1}{2} x_{0}^{2} \end{bmatrix} \end{equation*} $$
Resolviendo $$ \begin{align*} a_{0} &= \frac{1}{2} h \\ a_{1} &= \frac{1}{2} h \end{align*} $$
Reemplazando en (7) $$ \begin{equation*} \int_{x_{0}}^{x_{0}+h} f(x) \ dx = \frac{1}{2} h \ [f(x_{0}) + f(x_{0}+h)] \end{equation*} $$