Cfrgtkky

I have the following question and in the notes we are taught about general gaussian integration and gauss-legendre, but only briefly about gauss-laguerre so I am a bit stuck.

The question is as follows:

$$ text{Using the two point Gauss-laguerre integration method estimate:}$$
$$frac{1}{e^pi}int_{-pi}^{infty} (ye^{-y}sin(y) + pi e^{-y} sin(y))text{ }dy$$

From the notes I know that: $int_{0}^{infty}e^{-x}f(x) dx approx sum_{i=1}^{n}w_if(x_i)$ where $x_i$ is the i-th root of Laguerre polynomial $L_n(x).$

So what I have done so far is re arrange the original equation to get it into the format $int_{-pi}^{infty} e^{-y}f(y)dy$ and work out the roots of $L_2(x)$:

$$frac{1}{e^pi}int_{-pi}^{infty} e^{-y}(ysin(y) + pi sin(y))text{ }dy$$
$$L_{2}(x)=frac{x^2}{2}-2x+1$$ where the roots are: $$x_{0}=2-sqrt{2} text{ , } x_{1}=2+sqrt{2}$$

Not really sure where to go from here or even if I have started correctly. From putting the question into wolframalpha, I know that the value is -0.5.

Any help would be appreciated.