UnsolvedIntegral: What is the Integral of x^(x) dx?

[asd1249jf]

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.

 

[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 I've read at least one or two with the same question.

 

[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.

 

[neutrino]

Even Mathematica does not have a solution to this conundrum.

Mathematica could not find a formula for your integral. Most likely this means that no formula exists.

http://integrals.wolfram.com/

 

[Kurdt]

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.