1. Not finding help here? Sign up for a free 30min tutor trial with Chegg Tutors
    Dismiss Notice
Dismiss Notice
Join Physics Forums Today!
The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

Integral of 1/ln(x)

  1. Mar 26, 2008 #1

    rock.freak667

    User Avatar
    Homework Helper

    1. The problem statement, all variables and given/known data

    Find

    [tex]\int \frac{1}{lnx} dx[/tex]


    3. The attempt at a solution

    Let [itex]t=lnx \Rightarrow \frac{dt}{dx}=\frac{1}{x} \Rightarrow dx=e^t dt[/itex]



    [tex]\int \frac{1}{lnx} dx \equiv \int \frac{e^t}{t} dt[/tex]

    and well

    [tex]e^t= \sum _{n=o} ^{\infty} \frac{t^n}{n!}[/tex]

    [tex]\frac{e^t}{t}=\sum _{n=o} ^{\infty} \frac{t^{n-1}}{n!}[/tex]


    [tex]\int \frac{e^t}{t}=\int \sum _{n=o} ^{\infty} \frac{t^{n-1}}{n!}[/tex]

    [tex]=\sum _{n=o} ^{\infty} \frac{t^{n}}{(n+1)!}[/tex]

    Is there any easier closed form solution for this?
     
  2. jcsd
  3. Mar 26, 2008 #2
  4. Mar 26, 2008 #3
    Apparently it's.

    [tex]Ei\:(1,-1\ln(x))[/tex]

    This seems a little unfair though unless you knew about such an integral?
     
    Last edited: Mar 26, 2008
  5. Mar 26, 2008 #4

    rock.freak667

    User Avatar
    Homework Helper

    Nope,never even heard of it. But my usual way is, if you can't get it out directly, make a series and approximate and hopefully it might turn into something nicer.
     
  6. Mar 26, 2008 #5
    Probably but with the exponential integral and imaginary numbers there it will only be equivalent to the actual integral in a series. Although I think in that case they are asking you to show the series rather than the actual integral as it is in distinct maths language. Like to see how to do this, sorry but I have no idea, so can't help, I just thought the answer my maths program spat out was quite interesting, good luck.

    At least you know that the answer involves an exponentiation, and imaginary numbers I suppose.
     
    Last edited: Mar 26, 2008
  7. Mar 26, 2008 #6

    tiny-tim

    User Avatar
    Science Advisor
    Homework Helper

    erm … [tex]=\sum _{n=o} ^{\infty} \frac{t^{n}}{n.n!}\,.[/tex] :smile:
     
  8. Mar 26, 2008 #7

    rock.freak667

    User Avatar
    Homework Helper

    Ah silly me...forgot that n-1+1 is n and not n+1..thanks
     
Know someone interested in this topic? Share this thread via Reddit, Google+, Twitter, or Facebook

Have something to add?