Unusual Integral?

1. May 16, 2007

l46kok

Hello.

I was walking from home and thought to myself

What would the integral of x^(x) dx be?

And I momentarily thought

[X^(x+1)] / (x+1)

But that can't be it. lol

Does anyone have an idea? Oh just in case you guys become frantic and mad about it and go like (This is a homework problem, think about it yourself a little bit more ), I assure, you, this isn't a homework problem so please chill and give me some ideas.

2. May 16, 2007

trajan22

I'm pretty sure there is no integral for this that is defined by elementary functions. If you look around the forum I think Ive read at least one or two with the same question.

3. May 16, 2007

Dick

I thought that sounded familiar, too. A problem that's come up periodically is inverting the function x*e^x. This is the Lambert W function (non-elementary). But this is not that. It may have a name - but giving something a name doesn't mean you understand it better. It just makes it easier to google for it.

Last edited: May 16, 2007
4. May 17, 2007

neutrino

Even Mathematica does not have a solution to this conundrum.

http://integrals.wolfram.com/

5. May 17, 2007

Kurdt

Staff Emeritus
Of course its easier if you have these things in terms of the standard family of functions. For this particular case we recognise that its a "special" exponential function which can of course be written as follows.

$$x^x=e^{x\ln(x)}$$

Its actually relatively easy to differentiate but as yet I can't find any method of integrating, but perhaps someone with more experience would be able to find something.