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Integrating, probably by parts

  1. Oct 26, 2006 #1
    I have the expression [tex]\int{x(\ln{x})^3dx}[/tex]
    I thought I had a quick way to integrate by parts but it turned out that I had accidentally evaluated [tex]\int{x\ln{x}dx}[/tex] instead.
    Revisiting [tex]\int{x(\ln{x})^3dx}[/tex], I wanted to start by making a strange substitution, wherein u=ln(x), du=1/x dx, and x=e^u. This meant that when I rewrote the integral, instead of multiplying dx by a constant to get it to be du, I multiplied it by x (which in this case was e^u). Is that allowed? Because I got a very different, much uglier answer than the book's.

    I'd appreciate any comments, whether on my weird "method" or on a more standard approach to evaluating [tex]\int{x(\ln{x})^3dx}[/tex]
  2. jcsd
  3. Oct 26, 2006 #2
    Try integration by parts with u = (ln(x))^3 and dv = x dx
    Last edited: Oct 26, 2006
  4. Oct 26, 2006 #3
    Your substitution method should work fine. Your should be integrating [tex]\int{Exp[2u] u^3du}[/tex]. If you do it by integration by parts, you will need to do it 3 times.
  5. Oct 26, 2006 #4
    thanks, that got me the book's answer!
  6. Oct 27, 2006 #5
    And yes, the other way does work also. Nifty!
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