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Need help with integration by parts

  1. Dec 4, 2012 #1
    Hi all this is my first post hopefully i do it right.
    1. The problem statement, all variables and given/known data
    integrate ln^2(6x)dx



    3. The attempt at a solution
    *integral* ln^2(6x)dx
    u=ln^2(6x) dv=dx
    du=(2ln(6x))/x dx v=x

    xln^2(6x)-*integral*x(2ln(6x))/x dx

    xln^2(6x)-2*integral*ln(6x) dx

    u=ln(6x) dv=dx
    du=1/x dx v=x

    xln^2(6x)-2(xln(6x)-*integral*x(1/x)dx)
    xln^2(6x)-2(xln(6x)-*integral*dx)

    MY ANSWER.... that is not correct

    xln^2(6x)-2xln(6x)-2x+Constant

    i do not know where i am going wrong and i think im using parts correctly..i dont see any place where substitution could be used but i could be wrong
     
  2. jcsd
  3. Dec 4, 2012 #2

    Dick

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    Hah. Well done! I think the only problem is that the -2x should be +2x. You should be able to find where that mistake happened pretty easily.
     
  4. Dec 4, 2012 #3
    is it because the -2 is distributed and not +2?
     
  5. Dec 4, 2012 #4

    Dick

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    If you mean what I think, yes. xln^2(6x)-2(xln(6x)-*integral*dx). (-2)*(-1)=+2.
     
  6. Dec 4, 2012 #5
    yes! thank you so much.... those simple mistakes will be the death of me!
     
  7. Dec 5, 2012 #6

    SteamKing

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    In your u substitution, have you applied the chain rule correctly?
     
  8. Dec 5, 2012 #7
    Yes, I believe have.
    Using prime notation:

    ln^2(6x)'

    u=ln^2(6x)
    u'=2ln(6x)ln(6x)'
    u'=2ln(6x)/(6x)*(6x)'
    u'=2ln(6x)/(6x)*6
    since ((6x^1)' = *Const*x^n=*Const*nx^n-1 in my case 6x^1 = 1*6x^1-1
    the 6's cancel and you are left with
    u'=2ln(6x)/x :)
     
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