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Is there a closed form for this?

  1. Jan 25, 2016 #1
    Hi,

    I have this integral:

    [tex]\int_0^∞ \ln(1+x)\,e^{-x}\,dx[/tex]

    Is there any closed form expression for it?

    Thanks
     
  2. jcsd
  3. Jan 25, 2016 #2

    Samy_A

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    Integration by parts shows that the result contains the Exponential integral function Ei evaluated in -1.
    Depends of whether you call that a closed form expression.
     
  4. Jan 25, 2016 #3
    No, I need it in basic functions that have simple derivatives.
     
  5. Jan 25, 2016 #4

    Samy_A

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    Well, the derivative of Ei is ##\frac{e^x}{x}##.
     
  6. Jan 25, 2016 #5
    OK, thanks. Does ##e^{-x}## goes to zero faster than ##\ln(1+x)## goes to infility as ##x→∞##?
     
  7. Jan 25, 2016 #6

    Samy_A

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    Yes. If not your integral wouldn't give a finite result. ##e^{-x}## also goes to zero faster than any polynomial goes to infinity as ##x→+∞##.
     
    Last edited: Jan 25, 2016
  8. Jan 25, 2016 #7
    OK. I worked the analysis, and it won't work with me. I have something like this

    [tex]
    \int_0^∞ \ln(1+s\,x)\,e^{-x}\,dx
    [/tex]

    and I need to find

    [tex]
    \frac{\partial}{\partial s}\int_0^∞ \ln(1+s\,x)\,e^{-x}\,dx
    [/tex]

    from which I need to find ##s##. The integral is a complicated function of the argument ##s##.
     
  9. Jan 25, 2016 #8

    Krylov

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    See https://www.physicsforums.com/threads/derivative-of-integral.853211/

    As discussed there, you actually need to find the Fréchet derivative of the functional acting on the function s. I have made some comments on this in that topic, but I don't think you found them very helpful, so I will rest my case. However, I wanted to point out that this problem was already discussed.
     
  10. Jan 25, 2016 #9
    It is not that I found your comments unhelpful, but I just wanted to know enough to do the analysis, without going into the details. If I follow the same logic as the paper I attached there, and applied it to my problem here, I will have the p.d.f of ##x## appearing in the derivative. How to handle this when I need some numerical results?
     
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