# Integral of e^(x^2)*dx?

### #1

Posted 20 July 2007 - 12:20 AM

Anyway, I was making a math study the other day on integrals and things like that. You know, you get to the point where you study complex ways of integrating functions...

However, I wonder... is there an integral for the function mentioned above...

Can we integrate every function no matter how weird is it?

I tried integrating that equation:

e^(x^2)dx ... and man, that was frustating... you only end up repeating the same process again and again...

so, I ask... is there a solution to this one? If not, why not?

### #2

Posted 20 July 2007 - 02:32 AM

A way to see this is to start from the integral squared (change from x to -x, bounds do not change because you get one minus sign from the variable change and one for leaving the bounds as they are) and then to pass in polar coordinates:

[math](\int_{-\infty}^{\infty}e^{-x^2}dx)^2=(\int_{-\infty}^{\infty}e^{-x^2}dx)\cdot (\int_{-\infty}^{\infty}e^{-y^2}dy)=[/math]

[math](\int_{-\infty}^{\infty}e^{-(x^2+y^2)}dxdy)= \int_0^{2\pi}d\phi\int_0^\infty dr r e^{-r^2}=-\pi e^{-r^2}\vert^\infty_0=\pi[/math]

hence the integral is [imath]\sqrt{\pi}[/imath]

### #3 Guest_Lambus_*

Posted 20 July 2007 - 04:26 AM

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### #4

Posted 21 July 2007 - 04:21 AM

And the answer to your question is a resouding no, no we cannot integrate any function. The function e^(x^2) is of a class a functions who antiderivatives are defined as transcendental (neither elementary nor algebraic) and are not expressed in terms of a "normal" function. And the reason for there be no explicit formulae (as opposed to an infinite chain of polynomials) for certain integrals is rather complicated one, and I think I need some time to brood over that one (when it's not six in the morning). But interesting question, you'd never believe how many people get their doctorates in mathematics and never ask such simple questions.

And Sanctus, I've seen the derivation of this integral, but the one thing I never understood is why the integral squared is equal to the integral times the integral with respect to y. This is probably due to my unfamiliarity with multivariable calculus.

### #5

Posted 21 July 2007 - 04:45 AM

Also maybe the derivation of the integral the way I showed is complete only if you first show that the the definte integral not squared converges.

Hope you understand what I want to say.

### #6

Posted 22 July 2007 - 04:14 AM

### #7

Posted 22 July 2007 - 04:24 AM

### #8 Guest_Lambus_*

Posted 22 July 2007 - 04:53 AM

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### #9

Posted 22 July 2007 - 10:08 AM

Check it out Differential Galois theory - Wikipedia, the free encyclopedia

### #10 Guest_Lambus_*

Posted 23 July 2007 - 01:17 AM

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### #11

Posted 23 July 2007 - 03:21 AM

Actually, I'd say the computation of the square shows the integral to be convergent. It finds the integral in r to be, and a quantity that's neither defined nor finite couldn't have a finite defined square.Also maybe the derivation of the integral the way I showed is complete only if you first show that the the definte integral not squared converges.

### #12

Posted 23 July 2007 - 04:09 PM

### #13

Posted 02 February 2010 - 12:49 AM

I have actually meant to take time to look at functions of the form f(x) = x^m * e^(x^k), but I have yet to get around to that.

I'm not sure if you ever got round to doing this, but I think it can be integrated quite easily using a reduction formula as long as m is odd.

### #14

Posted 15 February 2010 - 12:21 PM

hence the integral is [imath]\sqrt{\pi}[/imath]

Hi!

You have made a nice proof. Later I have found similar proof here: Gaussian integral - Wikipedia, the free encyclopedia

But what about constructive proof of Normal Distribution probability density function:

???

I am assured all of us know about its properties and practical implementation.

But how did Carl Friedrich Gauss obtained this formulae?

Do You know any books or papers where this problem explained?

### #15

Posted 16 February 2010 - 07:36 AM

### #16

Posted 16 February 2010 - 07:46 PM

what do you mean by constructive proof?

**Sanctus**, I mean a proof like the proof stated here: Gauss_formulae_constructive_proof.djvu (7 pages)

Text is written in russian but You can easily understand it looking at figures & formulas.

The explanation of DJVU format is here: How to open .djvu files..........

Common idea of proof is usage of target shooting model...