gistfile1.txt

say we start with two linear functions

f(x) = a*x + b
g(x) = c*x + d

from previous work, we have their integrals (ignoring the offset)

F(x) = 1/2*a*x^2 + b*x 
G(x) = 1/2*c*x^2 + d*x

now we compare the integral of the product of f(x)*g(x)

j(x) = [f(x)*g(x)]dx = 1/3*a*c*x^3 + 1/2*(b*c + a*d)*x^2 + b*d*x

with the (much easier to compute) derivative of the product of F(x)*G(x)

k(x) = (F(x)*G(X))' = a*c*x^3 + 3/2*(b*c + a*d)*x^2 + 2*b*d*x

when we compare the coefficients of j(x) and k(x) by division, we notice that they relate like this:

1/2 * (2*b*d) = b*d
1/3 * (3/2*(b*c + a*d)) = 1/2*(b*c + a*d)
1/3 * (a*c) = a*c