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