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-   -   Integrate the following (https://www.askmehelpdesk.com/showthread.php?t=407303)

  • Oct 18, 2009, 11:34 AM
    Nhatkiem
    Integrate the following
    Last term, in my diff eq class during an exam and my professor made a mistake on one of the problems :rolleyes:. Anyway, the problem he gave us was pretty difficult to integrate.

    Integrate with respect to x: e^x/x

    Substitution and by parts didn't seem to work. The closest I got was to separate the expression as

    (1/x)(sum from n=1 to infinity of x^n/n!), I believe that is the correct expansion of e^x, I may be wrong.

    I'm just wondering if there is another approach to this problem.
  • Oct 18, 2009, 02:17 PM
    galactus
    Yes, this is a toughy. It is not soable by elementary means. That is why parts and

    substitution would not work. Your professor probably meant



    This has no closed form in elementary terms.

    Over the yesrs mathematicians have came up with forms in order to represent things like this that are hard to deal with.

    This one of yours is a form of the Exponential Integral, Ei(x), and can be represented by



    If you had limits, then we could probably evaluate. But an indefinite form is another matter.
  • Oct 18, 2009, 02:24 PM
    Nhatkiem

    Actually he meant x/e^x, which can be integrated by parts.

    But thank you non-the-less, I think I'm going to read up some more about this Ei(x) form!
  • Oct 18, 2009, 02:26 PM
    galactus
    Just Google the Exponential Integral. It is beyond elementary calculus and can be found in suh texts as Mathematical Physics.



    One can find the asymptotic series for the expoenntial integral or express it as an Incomplete Gamma function.

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