Is there a good approximation for this Gaussian-like integration? Is there an analytic solution or approximation for the following Gaussian-like integration? $\frac{1}{\eta^{2n}} \frac{1}{\sqrt{2 \pi}} \int_{-\eta}^{+\eta} e^{-x^2/2} x^{2n} dx$?  The numerical plot suggests that it initially decrease faster, but reach a steady decrease of $(2n)^{-1.06}$ numerically when $2n > 100$ for all $\eta$.

 A: An exact result (in terms of the incomplete Gamma function) and the large-$n$ asymptotics are as follows:
$$\frac{1}{\eta^{2n}} \frac{1}{\sqrt{2 \pi}} \int_{-\eta}^{+\eta} e^{-x^2/2} x^{2n} dx=\pi^{-1/2}2^{n}  \eta^{-2 n} \left[\Gamma \left(n+\tfrac{1}{2}\right)-\Gamma \left(n+\tfrac{1}{2},\tfrac{1}{2}\eta^2\right)\right]$$
$$\rightarrow \frac{e^{-\eta^2/2} \eta}{\sqrt{2 \pi } n},\;\;\text{for}\;\;n\gg 1.$$
The convergence to the large-$n$ result is shown in the plot for $\eta=5$ (blue is the integral, gold the large-$n$ value):

A: See paragraph 3.8 in https://people.sc.fsu.edu/~%20jburkardt/presentations/truncated_normal.pdf and https://people.smp.uq.edu.au/YoniNazarathy/teaching_projects/studentWork/EricOrjebin_TruncatedNormalMoments.pdf where  recursive Formulas are given for the moments of a truncated normal distribution.
A: Mathematica 12.0 answers
AsymptoticIntegrate[Exp[-x^2/2]*x^(2*n), {x, -\[Eta], \[Eta]}, {n, Infinity, 1},
Assumptions -> n>0&&n\[Element] Integers && \[Eta] > 0]/  Sqrt[2*Pi]/\[Eta]^(2*n)//Simplify

$$\frac{e^{-\frac{\eta ^2}{2}} \eta ^{-2 n} \left(\eta ^2\right)^{n+\frac{1}{2}}}{\sqrt{2 \pi } n} $$
