I need to solve integral using Gauss-Lagerre'e quadrature: $\int_0^\infty e^{-10x}(5x^3-\pi)$ on interval [0, $\infty$).
I am not really familiar with this kind of methods, but I need to solve it this way. This is how I am doing that, but I think everything is going bad.
I know that on the beginning I need to determinate the orthogonal n degree, where my n = 3:
$L_0(x)=1$
$L_1(x)=x-1$
$L_2(x)=x^2-4x+0$
$L_3(x)=x^3-9x^2+18x-6$
The weight function $p(x)$ that ensures convergence of integrals is $e^{-x}$.
Now I set the $A_k$ coefficient(whatever that means) using the expression below:
$A_k = \frac{((N+1)!)}{L_{N+1}(x_k)L_{N+2}(x_k)}$ ,
where $x_k$ are the real zeros of the polynomial.
At the end I have the integration expression:
$\int_0^\infty e^{-x}f(x)dx = \sum_{k=0}^N A_kf(x_k)$
So lets get started.
1)Calculations of the $A_k$.
$A_0 = \frac{((0+1)!)}{-1*(-1)} = \frac{1}{1} = 1$ //It's wrong because there is not real zero of $L_0(x)=1$ ?
$A_1 = \frac{2!}{1} = 2$
$A_2 = \frac{3!}{4*0} = \frac{6}{0} $ NEVER divide by 0!!!
etc.
What I am doing wrong? How to calculate integrals using given method?