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Strange Integration Problem

  1. Jun 12, 2006 #1
    A few of my friends and I have been trying to integrate/derive the following:

    f(x) = x^x

    without success. I'm not sure if it can be done conventionally, but I was wondering if anyone had any thoughts on this one. Thanks.
  2. jcsd
  3. Jun 12, 2006 #2


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    It cannot be done "conventionally", i.e. you cannot express its primitive in a closed form of elementary functions, which is also the case for e^(x²), sqrt(sin(x)), sin(x)/x, ...
  4. Jun 12, 2006 #3


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    While [tex]\int x^x dx[/tex] cannot be expressed in a finite number of elementary functions, derivative of [tex]f(x)= x^x[/tex] can be obtained by logarithmic differentiation: take the log of both sides to get

    [tex]ln[f(x)]=ln\left( x^x\right) = x ln(x)[/tex]

    now differentiate both sides to get

    [tex]\frac{f^{\prime}(x)}{f(x)}= ln(x)+1[/tex]

    multiply by f(x) to get

    [tex]f^{\prime}(x)= f(x)( ln(x)+1) = x^x( ln(x)+1)[/tex]
  5. Jun 13, 2006 #4
    Attached is a graph of [tex]y=\int_0^x u^u du[/tex], courtesy of Apple Grapher. :smile:

    Attached Files:

    Last edited: Jun 13, 2006
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