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Unusual Integral?

  1. May 16, 2007 #1
    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 :devil:), I assure, you, this isn't a homework problem so please chill and give me some ideas.
     
  2. jcsd
  3. May 16, 2007 #2
    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.
     
  4. May 16, 2007 #3

    Dick

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    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
  5. May 17, 2007 #4
    Even Mathematica does not have a solution to this conundrum. :biggrin:

    http://integrals.wolfram.com/
     
  6. May 17, 2007 #5

    Kurdt

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    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.

    [tex]x^x=e^{x\ln(x)}[/tex]

    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.
     
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