Boole's rule

In mathematics, Boole's rule, named after George Boole, is a method of numerical integration. It approximates an integral

\int_{x_1}^{x_5} f(x)\,dx

by using the values of ƒ at five equally spaced points

x_1, \quad x_2 = x_1 + h, \quad x_3 = x_1 + 2h, \quad x_4 = x_1 + 3h, \quad x_5 = x_1 +4h. \,

It is expressed thus in Abramowitz and Stegun (1972, p. 886):

\int_{x_1}^{x_5} f(x)\,dx = \frac{2 h}{45}\left( 7f(x_1) + 32 f(x_2) + 12 f(x_3) + 32 f(x_4) + 7f(x_5) \right) + \text{error term},

and the error term is

-\,\frac{8}{945} h^7 f^{(6)}(c)

for some number c between x₁ and x₅. (945 = 1 × 3 × 5 × 7 × 9.)

It is often known as Bode's rule, due to a typographical error that propagated; e.g. in Abramowitz and Stegun (1972, p. 886).^[1]

References