Boole's rule
= Boole's rule
{c}
{title2=$\int_{-1}^1p=(7p(-1)+32p(-1/2)+12p(0)+32p(1/2)+7p(1))/45$}
Integrate the five <Lagrange interpolation polynomials> at the equally spaced nodes to obtain the displayed rule. It is exact for <polynomials> through degree four, and symmetry also makes it exact in degree five. Symmetric endpoint, inner-node and midpoint weights $a,b,c$ satisfy $2a+2b+c=2$, $2a+b/2=2/3$, $2a+b/8=2/5$, giving $a=7/45$, $b=32/45$, $c=12/45$.