I am looking for the best formula for integrating the following:
$$ \int_{0}^{\infty} e^{-x^2}P_5(x) $$
Where $P_5(x)$ is a generic polynomial of degree 5.
I was thinking of using Hermite formula, but it only works with the boundary of the integral between $-\infty$ and $+\infty$, I tried to play around a bit with integration extrema but without avail.
I also tried to use Laguerre integrals but the problem is that if I fix the $x^2$ I have a square root inside the polynomial!
Could I have a hint?
By parts, for $k\ge2$, $$I_k:=\int_0^\infty x^ke^{-x^2}dx=-\left.\frac{x^{k-1}}2e^{-x^2}\right|_0^\infty+\frac{k-1}2\int_0^\infty x^{k-2}e^{-x^2}dx=\frac{k-1}2I_{k-2}.$$
Then
$$I_1=-\left.\frac{e^{-x^2}}2\right|_0^\infty=\frac12$$ and
$$I_0=\frac{\sqrt\pi}2$$ by the known formula.
This is enough to integrate for any polynomial.