Calculate the following improper integrals
$ \displaystyle{ \int_{e}^{\infty} e^{-\frac{1}{2} (nx)^2} dx , \quad \int_{e}^{\infty}x^2 e^{-\frac{1}{2} (nx)^2} dx \quad ,\int_{-\infty}^{\infty} xe^{-\frac{1}{2} (nx)^2} dx }$.
I know that $ \displaystyle{ \int_{0}^{\infty} e^{-a x^2} dx =\frac{1}{2} \sqrt{\frac{\pi}{2}} }$ where $ a>0$
I have also read that $ \displaystyle{ \int_{0}^{\infty} e^{-a x^2} x^{2n}dx= \frac{1 \cdot 3 \cdot 5 \cdot \cdots \cdot (2n-1)}{2^{n+1} a^n} \sqrt{ \frac{\pi}{2}} }$ where $ n \in \mathbb N$.
So I think that for the first it is enough to compute $ \displaystyle{ \int_{0}^{e} e^{-\frac{1}{2} (nx)^2} dx } $ and for the second it is enough to compute $\displaystyle{\int_{0}^{e} x^{2} e^{-\frac{1}{2} (nx)^2} dx }$.
But I have no idea from here on.
Is there another approach which works better ?
I would really appreciate some help on any of the above integrals.
Thank's in advance!
edit: Sorry for putting the three integrals in one question but it seems to me that there is a connection between them ( on how I calculate them) since they are similar.
edit2: Can someone give me a proof or a link to see how can I get the series in the link http://people.math.sfu.ca/~cbm/aands/page_932.htm which was given in the answer below.
Thank's!