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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!

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    @passenger $u=-\frac{1}{2}(nx)^2$.2012-05-27

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There is no solution in a finite number of terms involving the elementary functions for the first integral; its value is $a(1-\Phi(y))$ where the values of $a$ and $y$ can be determined via a change of variables that transform the integral into the integral formula for $aP\{X > y\}$ where $X$ is a standard normal random variable.

The second integral should be integrated by parts to simplify it a little. Try going the other way first: what is the derivative of $e^{-n^2x^2/2}$? Can you express the integral you need to compute in terms of $\int x\ \mathrm d(e^{-n^2x^2/2})$ and use integration by parts?

The third question you have solved already as your response to Nate Eldredge's hint seemed to indicate, but if not, look again to my suggestion of the derivative of $e^{-n^2x^2/2}$.

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    See [this link](http://people.math.sfu.ca/~cbm/aands/page_932.htm)2012-05-27