Gauss quadrature infinite segment 
Given a weighted integral $$I(f)=\int_{-\infty}^\infty f(x)e^{-x^2}dx.$$ How can I calculate the Gauss quadrature for two points.

I know how to calculate the quadrature wih Legendre polynomials, but here it does not fit, because of the bounds of the integral.
 A: Gauss-Legendre
Gauss-Legendre quadrature is designed for integrals over the integration domain
$[−1,+1]$, but this can be modified to accept any finite domain $[a, b]$ through the transformation
$$\int_{a}^{b}f(x)dx=\frac{b-a}{2}\int_{-1}^{1}f\left(\frac{b-a}{2}x+\frac{b+a}{2}\right)dx$$
so $a=-\infty$ and $b=\infty$, we must truncate the upper and lower limits:
$$\int_{x_{\min}}^{x_{\max}}f(x)dx=\frac{x_{\max}-x_{\min}}{2}\int_{-1}^{1}f\left(\frac{x_{\max}-x_{\min}}{2}x+\frac{x_{\max}+x_{\min}}{2}\right)dx$$
Gauss-Laguerre Quadrature
$$I(f)=\int_{-\infty}^{0} f(x){e^{-x^2}}dx+\int_{0}^{\infty} f(x){e^{-x^2}}dx=\int_{0}^{\infty} f(-x){e^{-x^2}}dx+\int_{0}^{\infty} f(x){e^{-x^2}}dx$$
Gauss-Laguerre quadrature is especially relevant for the purposes of evaluating the integral, because it is designed for integrals over the integration domain $(0,\infty)$. Suppose we wish to apply Gauss-Laguerre quadrature with N points. The abscissas $(x_1, . . . , x_N)$ are the roots of the Laguerre polynomial $L_N(x)$ of order $N$ defined as
$${{L}_{N}}(x)=\sum\limits_{k=0}^{N}{\frac{{{(-1)}^{k}}}{k!}}\left( \begin{matrix}
   N  \\
   k  \\
\end{matrix} \right){{x}^{k}}
$$
The weights
$(w_1, . . . ,w_N)$ are obtained with the derivative of $L_N(x)$ evaluated at each of the $N$ abscissas
$${{w}_{j}}=\frac{(N!)^2{{e}^{{{x}_{j}}}}}{{{x}_{j}}{{(L{_{N}'}({{x}_{j}}))}^{2}}}\,\,,\,\,j=1,2,\ldots ,N$$
