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What is the integration of (ln(x))^n?

  1. Dec 9, 2009 #1
    what is the integration of (ln(x))^n?
     
  2. jcsd
  3. Dec 9, 2009 #2

    HallsofIvy

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    Science Advisor

    Re: integral

    To integrate ln x by parts, let u= ln x, dv= dx. Then du= (1/x)dx and v= x so
    [tex]\int ln x= x(ln x)- \int dx= xln x- x+ C[/tex]

    To integrate [itex](ln x)^2[/itex] by parts, let u= ln x, dv= ln x dx. Then du= (1/x)dx and v= xln x- x so
    [tex]x(ln x)^2- x ln x- \int ln x- 1 dx= x(ln x)^2- x ln x- (x ln x - x- x)= x(ln x)^2- 2x+ C[/tex]

    Keep integrating by parts until you see a pattern.
     
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