How to find x's for representing a specific type of integral using a quadrature formula?

Question

I have this integral which I want to approx with quadrature formula for some fixed $n$: $$ \int\limits_0^{+\infty} x e^{-x} f(x) dx \approx \sum\limits_{k = 0}^n A_k f(x_k) $$

I found info about how to find $A_k$: $$ A_k = \dfrac{n!\Gamma(\alpha + n + 1)}{x_k\left(L_n'^{(\alpha)}(x_k)\right)^2}, $$ when $\Gamma(x)$ - gamma-function and $L_n^{(\alpha)}(x)$ - Laguerre polynom.

But how to find $x_k$?

Answer 1

This is a Gauss-Laguerre quadrature, your $x_k$ are the roots of Laguerre polynomials of corresponding degree $n$.