I need to calculate a quadrature rule with maximum degree of accuracy that looks like this:
$ \int_0^\infty e^{-x}f(x)dx = \sum_{i=0}^n A_if(x_i) + R_n(f) $
where $n=2$.
For $R_n(f)$ I have this formula:
$ R_n(f) = \frac{f^{(2n)}(\xi)}{(2n)!} \, (\pi_n,\pi_n) $
I've already calculated $A_n$ and $x_n$ and I already know that $\pi_n$ is a Laguerre polynomial.
As joriki pointed out, Laguerre polynomials are orthogonal so I'm left with this:
$ R_2(f) = \frac{f^{(2n)}(\xi)}{24} $
My question is: how do I choose $\xi$? Or do I just leave it like that?