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Integrate x^x

  1. Sep 30, 2005 #1
    Hello, a question: is there a reasonable way to obtain [tex]\int x^xdx[/tex] ??
  2. jcsd
  3. Sep 30, 2005 #2
    Not that my TI-89 knows of.
  4. Sep 30, 2005 #3


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    The integral cannot be expressed in elementary functions.
    If an infinite series will do
    [tex]\int \ x^xdx=\int \ e^{x\log(x)}dx=C+\sum_{k=0}^\infty \ \int_0^x \ \frac{\log^k(t)}{k!}t^kdt[/tex]
    so if an infinite sum will do find an easy integral gets you a nice one.
    [tex]\int \ x^{-x}dx[/tex]
    is similar
    [tex]\int_0^1 x^xdx[/tex]
    [tex]\int_0^1 x^{-x}dx[/tex]
    are extra nice
    Last edited: Sep 30, 2005
  5. Oct 2, 2005 #4
    Thanks to both of you.

  6. Oct 4, 2005 #5


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    Yea thanks.

    Is there a way to remove the integral sign? Looks like they can be analytically determined. for example:

    [tex]\int x^2ln^2(x)dx=2/3 x^3-2/9 x^3ln(x)+1/3 x^3ln^2(x)[/tex]

    and higher powers involve corresponding higher powers of x and ln(x) in the antiderivative.
    Last edited: Oct 4, 2005
  7. Oct 20, 2005 #6


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    It is provably impossible to represent that antiderivative as a finite combination of elementary functions. See the bottom of http://mathworld.wolfram.com/Integral.html for a confrimation of this fact.

    This is not to say that it does not have a solution, it just not pretty.
  8. Oct 20, 2005 #7
    The Problem could be solved by using the
    integral of the infinite series
    which could be calculated by the numerical methods
    if u are interested in the solution i may work out the
    algorithm or program for You
  9. Oct 20, 2005 #8

    My intuition tells me you can use the Lambert-W function on this one. Just as Eisenstein made it work for "power tower" functions (N^N^N^N^N^N^N...). It might work.

    If you want to know about that function, check the link on the post "A very interesting question about Complex Variable"
  10. Oct 30, 2005 #9
    Something tells me that might make it worse.
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