Cuadratura de Gauss

\begin{equation} \sum_{i=0}^{n} a_{i} \ f(x_{i}) = \int_{-1}^{1} f(x) \ dx \end{equation}

Fórmula de dos puntos

\begin{equation*} a_{0} f(x_{0}) + a_{1} f(x_{1}) = \int_{-1}^{1} f(x) \ dx \end{equation*}

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*}

Fórmula de tres puntos

\begin{equation*} a_{0} f(x_{0}) + a_{1} f(x_{1}) + a_{2} f(x_{2}) = \int_{-1}^{1} f(x) \ dx \end{equation*}

Usando $f(x) = 1$

\begin{equation*} a_{0} f(x_{0}) + a_{1} f(x_{1}) + a_{2} f(x_{2}) = \int_{-1}^{1} 1 \ dx \end{equation*}

Reemplazando valores e integrando

\begin{equation*} a_{0} + a_{1} + a_{2} = 2 \end{equation*}

Usando $f(x) = x$

\begin{equation*} a_{0} f(x_{0}) + a_{1} f(x_{1}) + a_{2} f(x_{2}) = \int_{-1}^{1} x \ dx \end{equation*}

Reemplazando valores e integrando

\begin{equation*} a_{0} x_{0} + a_{1} x_{1} + a_{2} x_{2} = 0 \end{equation*}

Usando $f(x) = x^{2}$

\begin{equation*} a_{0} f(x_{0}) + a_{1} f(x_{1}) + a_{2} f(x_{2}) = \int_{-1}^{1} 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} = \frac{2}{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}) = \int_{-1}^{1} 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} = 0 \end{equation*}

Usando $f(x) = x^{4}$

\begin{equation*} a_{0} f(x_{0}) + a_{1} f(x_{1}) + a_{2} f(x_{2}) = \int_{-1}^{1} 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} = \frac{2}{5} \end{equation*}

Usando $f(x) = x^{5}$

\begin{equation*} a_{0} f(x_{0}) + a_{1} f(x_{1}) + a_{2} f(x_{2}) = \int_{-1}^{1} x^{5} \ dx \end{equation*}

Reemplazando valores e integrando

\begin{equation*} a_{0} x_{0}^{5} + a_{1} x_{1}^{5} + a_{2} x_{2}^{5} = 0 \end{equation*}

Formando un sistema de ecuaciones

\begin{align*} a_{0} + a_{1} + a_{2} &= 2 \\ a_{0} x_{0} + a_{1} x_{1} + a_{2} x_{2} &= 0 \\ a_{0} x_{0}^{2} + a_{1} x_{1}^{2} + a_{2} x_{2}^{2} &= \frac{2}{3} \\ a_{0} x_{0}^{3} + a_{1} x_{1}^{3} + a_{2} x_{2}^{3} &= 0 \\ a_{0} x_{0}^{4} + a_{1} x_{1}^{4} + a_{2} x_{2}^{4} &= \frac{2}{5} \\ a_{0} x_{0}^{5} + a_{1} x_{1}^{5} + a_{2} x_{2}^{5} &= 0 \end{align*}

Resolviendo

\begin{align*} a_{0} &= \frac{5}{9} \\ a_{1} &= \frac{8}{9} \\ a_{2} &= \frac{5}{9} \\ x_{0} &= -\frac{\sqrt{15}}{5} \\ x_{1} &= 0 \\ x_{0} &= \frac{\sqrt{15}}{5} \end{align*}