Dismiss Notice
Join Physics Forums Today!
The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

How does one go about proving an elementary solution to an integral does not exist?

  1. Aug 12, 2009 #1
    Recently I have begun thinking about the function x^x. I am well aware that there is no elementary function to define it's antiderivative, and intuitively it makes sense (I cannot think of an elementary function who's derivative is x^x). However, how would one go about proving this rigorously?
     
  2. jcsd
  3. Aug 12, 2009 #2
    Re: How does one go about proving an elementary solution to an integral does not exis

    MARCHISOTO and ZAKERI (1994): "An Invitation to Integration in Finite Terms" , The College Mathematics Journal 25 No. 4 Sept. pp 295 - 308

    J. F. Ritt, Integration in finite terms: Liouville's theory of elementary methods, 1948
     
  4. Aug 12, 2009 #3
    Re: How does one go about proving an elementary solution to an integral does not exis

    Does anyone happen to have a more accessible resource about this subject? Like something online?
     
  5. Aug 13, 2009 #4
  6. Aug 13, 2009 #5
    Re: How does one go about proving an elementary solution to an integral does not exis

    so what we have is:

    [tex]
    \int f(x)e^g^(^x^)dx
    [/tex]
    [tex]
    g(x)=0
    [/tex]
    [tex]
    f(x)=x^x
    [/tex]

    so the formula given

    [tex]

    f(x)=R'(x)+g(x)R(x)

    [/tex]

    just goes to

    [tex]
    f(x)=R'(x)
    [/tex]

    which isn't overly illuminating :(
     
  7. Aug 13, 2009 #6
    Re: How does one go about proving an elementary solution to an integral does not exis

    No, [itex]x^x[/itex] is not a rational function, so this is not the way to fit the theorem.
    In fact, the case of [itex]x^x[/itex] (as explained in the actual references) is a bit more involved than that simplistic web page says.
     
  8. Aug 13, 2009 #7
    Re: How does one go about proving an elementary solution to an integral does not exis

    How about letting [tex] x^x = e^{x\ln x} [/tex]?
     
Know someone interested in this topic? Share this thread via Reddit, Google+, Twitter, or Facebook




Similar Discussions: How does one go about proving an elementary solution to an integral does not exist?
Loading...