# Método de los coeficientes indeterminados

$$\int_{x_{0}}^{x_{n}} f(x) \ dx \approx \sum_{i=0}^{n} a_{i} f(x_{i})$$

## Regla del trapecio

$$\int_{x_{0}}^{x_{1}} f(x) \ dx = a_{0} f(x_{0}) + a_{1} f(x_{1}) \label{ecuacion2}$$

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 \eqref{ecuacion2}

\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*}

## Regla de Simpson 1/3

$$\int_{x_{0}}^{x_{2}} f(x) \ dx = a_{0} f(x_{0}) + a_{1} f(x_{1}) + a_{2} f(x_{2}) \label{ecuacion3}$$

Usando $f(x) = 1$

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

Reemplazando valores e integrando

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

Reemplazando valores e integrando

\begin{equation*} a_{0} x_{0} + a_{1} x_{1} + a_{2} x_{2} = \frac{1}{2} x_{2}^{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}) = \int_{x_{0}}^{x_{2}} 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{1}{3} x_{2}^{3} - \frac{1}{3} x_{0}^{3} \end{equation*}

Formando un sistema de ecuaciones

\begin{equation*} \begin{bmatrix} 1 & 1 & 1 \\ x_{0} & x_{1} & x_{2} \\ x_{0}^{2} & x_{1}^{2} & x_{2}^{2} \end{bmatrix} \begin{bmatrix} a_{0} \\ a_{1} \\ a_{2} \end{bmatrix} = \begin{bmatrix} x_{2} - x_{0} \\ \frac{1}{2} x_{2}^{2} - \frac{1}{2} x_{0}^{2} \\ \frac{1}{3} x_{2}^{3} - \frac{1}{3} x_{0}^{3} \end{bmatrix} \end{equation*}

Resolviendo

\begin{align*} a_{0} &= \frac{2 x_{0}^{2} - 3 x_{0} x_{1} - x_{0} x_{2} + 3 x_{1} x_{2} - x_{2}^{2}}{6 (x_{1} - x_{0})} \\ a_{1} &= -\frac{x_{0}^{3} - 3 x_{0}^{2} x_{2} + 3 x_{0} x_{2}^{2} - x_{2}^{3}}{6 (x_{2} - x_{1}) (x_{1} - x_{0})} \\ a_{2} &= -\frac{x_{0}^{2} - 3 x_{0} x_{1} + x_{0} x_{2} + 3 x_{1} x_{2} - 2 x_{2}^{2}}{6 (x_{2} - x_{1})} \end{align*}

Reemplazando en \eqref{ecuacion3}

\begin{equation*} \int_{x_{0}}^{x_{2}} f(x) \ dx = \biggl[ \frac{2 x_{0}^{2} - 3 x_{0} x_{1} - x_{0} x_{2} + 3 x_{1} x_{2} - x_{2}^{2}}{6 (x_{1} - x_{0})} \biggr] f(x_{0}) - \biggl[ \frac{x_{0}^{3} - 3 x_{0}^{2} x_{2} + 3 x_{0} x_{2}^{2} - x_{2}^{3}}{6 (x_{2} - x_{1}) (x_{1} - x_{0})} \biggr] f(x_{1}) - \biggl[ \frac{x_{0}^{2} - 3 x_{0} x_{1} + x_{0} x_{2} + 3 x_{1} x_{2} - 2 x_{2}^{2}}{6 (x_{2} - x_{1})} \biggr] f(x_{2}) \end{equation*}

## Regla de Simpson 3/8

$$\int_{x_{0}}^{x_{3}} f(x) \ dx = a_{0} f(x_{0}) + a_{1} f(x_{1}) + a_{2} f(x_{2}) + a_{3} f(x_{3})$$

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}) = \int_{x_{0}}^{x_{3}} 1 \ dx \end{equation*}

Reemplazando valores e integrando

\begin{equation*} a_{0} + a_{1} + a_{2} + a_{3} = x_{3} - 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}) = \int_{x_{0}}^{x_{3}} 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} = \frac{1}{2} x_{3}^{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}) = \int_{x_{0}}^{x_{3}} 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} = \frac{1}{3} x_{3}^{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}) = \int_{x_{0}}^{x_{3}} 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} = \frac{1}{4} x_{3}^{4} - \frac{1}{4} x_{0}^{4} \end{equation*}

Formando un sistema de ecuaciones

\begin{equation*} \begin{bmatrix} 1 & 1 & 1 & 1 \\ x_{0} & x_{1} & x_{2} & x_{3} \\ x_{0}^{2} & x_{1}^{2} & x_{2}^{2} & x_{3}^{2} \\ x_{0}^{3} & x_{1}^{3} & x_{2}^{3} & x_{3}^{3} \end{bmatrix} \begin{bmatrix} a_{0} \\ a_{1} \\ a_{2} \\ a_{3} \end{bmatrix} = \begin{bmatrix} x_{3} - x_{0} \\ \frac{1}{2} x_{3}^{2} - \frac{1}{2} x_{0}^{2} \\ \frac{1}{3} x_{3}^{3} - \frac{1}{3} x_{0}^{3} \\ \frac{1}{4} x_{3}^{4} - \frac{1}{4} x_{0}^{4} \end{bmatrix} \end{equation*}

Resolviendo

\begin{align*} a_{0} &= -\frac{3 x_{0}^{3} - 4 x_{0}^{2} x_{1} - 4 x_{0}^{2} x_{2} - x_{0}^{2} x_{3} + 6 x_{0} x_{1} x_{2} + 2 x_{0} x_{1} x_{3} + 2 x_{0} x_{2} x_{3} - x_{0} x_{3}^{2} - 6 x_{1} x_{2} x_{3} + 2 x_{1} x_{3}^{2} + 2 x_{2} x_{3}^{2} - x_{3}^{3}}{12 (x_{2} - x_{0})(x_{1} - x_{0})} \\ a_{1} &= \frac{x_{0}^{4} - 2 x_{0}^{3} x_{2} - 2 x_{0}^{3} x_{3} + 6 x_{0}^{2} x_{2} x_{3} - 6 x_{0} x_{2} x_{3}^{2} + 2 x_{0} x_{3}^{3} + 2 x_{2} x_{3}^{3} - x_{3}^{4}}{12(x_{1} - x_{0})(x_{3} - x_{1})(x_{2} - x_{1})} \\ a_{2} &= \frac{x_{0}^{4} - 2 x_{0}^{3} x_{1} - 2 x_{0}^{3} x_{3} + 6 x_{0}^{2} x_{1} x_{3} - 6 x_{0} x_{1} x_{3}^{2} + 2 x_{0} x_{3}^{3} + 2 x_{1} x_{3}^{3} - x_{3}^{4}}{12(x_{0} x_{1} - x_{0} x_{2} - x_{1} x_{2} + x_{2}^{2})(x_{2} - x_{3})} \\ a_{3} &= -\frac{x_{0}^{3}- 2 x_{0}^{2} x_{1} - 2 x_{0}^{2} x_{2} + x_{0}^{2} x_{3} + 6 x_{0} x_{1} x_{2} - 2 x_{0} x_{1} x_{3} - 2 x_{0} x_{2} x_{3} + x_{0} x_{3}^{2} - 6 x_{1} x_{2} x_{3} + 4 x_{1} x_{3}^{2} + 4 x_{2} x_{3}^{2} - 3 x_{3}^{3}}{12(x_{1} x_{2} - x_{1} x_{3} - x_{2} x_{3} + x_{3}^{2})} \end{align*}

## Regla de Boole

$$\int_{x_{0}}^{x_{4}} f(x) \ dx = a_{0} f(x_{0}) + a_{1} f(x_{1}) + a_{2} f(x_{2}) + a_{3} f(x_{3}) + a_{4} f(x_{4})$$

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*}