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?
So, to answer my own question, you're not supposed to calculate $\xi$. The remainder term is just a way to gauge how big the approximation error is.
For example, if $f(x)=cos(x)$, the remainder is:
$$ R = \frac{cos(\xi)}{24} $$
which is not so bad, since $cos(x) \in [-1, 1]$.