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Never ending integration by parts

  1. Oct 12, 2011 #1
    1. The problem statement, all variables and given/known data
    [tex] \int_0^\infty{ \frac{1}{x} e^{-x}} [/tex]

    2. Relevant equations
    Integration by parts
    [tex] \int{u dv} = uv - \int{v du} [/tex]

    3. The attempt at a solution
    [tex] u = \frac{1}{x} [/tex]
    [tex] du = \frac{1}{x^2} dx [/tex]
    [tex] v = -e^{-x} [/tex]
    [tex] dv = e^{-x} dx [/tex]

    [tex] -\frac{1}{x} e^{-x} - \int_0^\infty{-e^{-x} \frac{1}{x^2}} dx [/tex]
    It looks like this process is going to go on forever because I can't get rid of the 1/x term. Could someone please give some guidance on how this is done? Thanks.
  2. jcsd
  3. Oct 12, 2011 #2


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    You can't reduce it to a simple function using integration by parts. The integral defines a special function called the 'exponential integral', Ei(x).
  4. Oct 12, 2011 #3
    Actually, the exponential integral Ei(x) is defined a little differently. See http://mathworld.wolfram.com/ExponentialIntegral.html for example. In any case, given your limits of zero to infinity, the integral diverges.
  5. Oct 12, 2011 #4


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    Ooops. Thanks for the correction. I didn't pay any attention to the fact it was a definite integral.
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