Regla del Trapecio
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{\displaystyle I = \int_{a}^{b} f(x) \, dx \approx (b-a) \frac{f(a)+f(b)}{2}}
Regla de trapecio de aplicación múltiple
hay n+1 puntos igualmente espaciados
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{\displaystyle x_0, x_1, x_2, ...,x_n}
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{\displaystyle x_0=a_1 \qquad x_n=b}
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{\displaystyle I = \int_{a}^{b} f(x) \, dx = \int_{x_0}^{x_1} f(x) \, dx + \int_{x_1}^{x_2} f(x) \, dx + ... + \int_{x_{n-1}}^{x_n} f(x) \, dx }
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{\displaystyle \qquad \approx h \frac{f(x_0)+f(x_1)}{2} + h \frac{f(x_1)+f(x_2)}{2} + h \frac{f(x_{n-1})+f(x_n)}{2}}
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{\displaystyle = \frac{h}{2} \left[f(x_0)+2\sum_{i=1}^{n-1} f(x_i) + f(x_n)\right] ;\qquad \qquad h= \frac{b-a}{n} }
Regla de Simpon
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{\displaystyle \int_{a}^{b} f(x) \, dx \approx \frac{b-a}{6}[(x_0)+4f(x_1)+f(x_2)]}
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{\displaystyle x_0=a,\quad x_1=\frac{a+b}{2},\quad x_2=b}
Regla de Simpson de aplicación múltiple con nodos equidistantes
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{\displaystyle a=x_0 < x_1 < x_2 < ... < x_n = b }
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{\displaystyle I = \int_{a}^{b} f(x) \, dx = \int_{x_0}^{x_2} f(x) \, dx + \int_{x_2}^{x_4} f(x) \, dx + ... + \int_{x_{n-2}}^{x_n} f(x) \, dx }
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{\displaystyle \qquad \approx (x_2-x_0) \frac{f(x_0)+4f(x_1)+f(x_2)}{6}\; +\; (x_4 - x_2) \frac{f(x_2)+4f(x_3)+f(x_4)}{6} \; +\; ...\; + \;(x_n-x_{n-2})\frac{f(x_{n-2})+4f(x_1)+f(x_n)}{6}}
<math>= \frac{2(b-a)}{6n} \left[f(x_0)+4\sum_{i=1,3,5}^{n-1} f(x_i) + \sum_{i=2,4,6}^{n-2} f(x_i) + f(x_n)\right]