Boole's rule

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

${\displaystyle \int _{x_{1}}^{x_{5}}f(x)\,dx}$

by using the values of ƒ at five equally spaced points

${\displaystyle x_{1},\quad x_{2}=x_{1}+h,\quad x_{3}=x_{1}+2h,\quad x_{4}=x_{1}+3h,\quad x_{5}=x_{1}+4h.\,}$

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

${\displaystyle \int _{x_{1}}^{x_{5}}f(x)\,dx={\frac {2h}{45}}\left(7f(x_{1})+32f(x_{2})+12f(x_{3})+32f(x_{4})+7f(x_{5})\right)+{\text{error term}},}$

and the error term is

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

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 from Abramowitz and Stegun (1972, p. 886).[1][2]