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Integration by parts x5(lnx)2 dx

  1. Mar 20, 2009 #1
    1. The problem statement, all variables and given/known data

    Integrate the following:
    (Ill use { as the integration sign)

    {x5(lnx)2 dx

    2. Relevant equations
    {u dv = uv - {v du

    I'm really new to integration by parts, and unfortunately im having to learn it out of a book for now. I sort of get the idea, but this one just doesn't look so right when im done with it.

    3. The attempt at a solution
    u= (lnx)2
    du = 2lnx(1/x) dx
    v = x6/6
    dv = x5dx

    {u dv = (x6/6)((lnx)2) - { (x6/6)(2 lnx)(1/x)dx
    = (x6/6)((lnx)2) - (1/3) { (x5)(lnx) dx

    At this point I would doing integration by parts again for the integral: { (x5)(lnx) dx.

    So:
    u = lnx du = 1/x dx dv = x5dx v = x6/6
    --> { (x5)(lnx) dx = (lnx)(x6/6) - { (x6/6)(1/x) dx
    = (lnx)(x6/6) - (1/6) { x5 dx
    = (lnx)(x6/6) - x6/12

    Plugging this back into my original solution:

    = (x6/6)((lnx)2) - (1/3)((lnx)(x6/6) - x6/12)
    =(x6/6)((lnx)2) - (lnx x6/18) - x6/36


    This could even further be simplified i suppose by taking out some common factors of x6/6, but I don't even know if all this is right. Some let me know if im on the right track, or where i went wrong??
     
    Last edited: Mar 20, 2009
  2. jcsd
  3. Mar 20, 2009 #2

    lanedance

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    hi LunarJK

    why not try and differentiate it and see if you get the original integrand?
     
    Last edited: Mar 20, 2009
  4. Mar 20, 2009 #3
    Well i tried... and i do not get the original integrand. I need some advice as to where i went wrong.
     
  5. Mar 20, 2009 #4

    lanedance

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    your method looks good I think you may have just mixed up some +- signs and factors

    i get to

    [tex]= \frac{1}{6}(x^6(\ln{x})^2 - \frac{1}{3}( x^6\ln{x} -\frac{x^6}{6}) )+c = \frac{x^6}{6.6.3}(6.\ln{x)((3.\ln{x} - 1) + 1 ))+c[/tex]

    which agrees with the online integrator
    http://integrals.wolfram.com/
     
  6. Mar 20, 2009 #5

    lanedance

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    this looks good here
    i think your 12 should be 6.6 = 36
    this should be -(-x^6/36) = +x^6/36 (i haven;t factored in the above error as well though...)
    hopefully this gets you there
     
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