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Homework Help: Integrating 1/xln(x) using integration by parts

  1. Sep 13, 2010 #1
    1. The problem statement, all variables and given/known data
    Integrating 1/xlnx by parts...
    2. Relevant equations
    Find the integral of 1/xlnx

    The question asks to solve by substitution, which I can do and results in ln(ln(x)) + c

    It then asks to compute using integration by parts, and then to explain how it can be true (because it will compute something different to substitution).

    3. The attempt at a solution
    I = uv - int (v dU)

    let u= 1/lnx du = 1/x(lnx)^2
    let dv = 1/x, v = lnx

    Sub into the parts formula

    I = lnx* 1/lnx - int (lnx/x(lnx)^2)
    I = lnx/lnx - int (1/xlnx) <--- what we started with
    I = 1 - int (1/xlnx) This is 1 - I, the integral we began with...

    I've bene shown this trick where you can go..
    I = 1- I
    2I = 1
    I = 1/2

    I'm not sure if this is correct, but I would appreciate any help

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

    2. Relevant equations

    3. The attempt at a solution
  2. jcsd
  3. Sep 13, 2010 #2
    It is correct
  4. Sep 13, 2010 #3
    Thank you for the quick reply. I'm also asked to explain how I = 1/2 can be true, when using substitution yields ln(ln(x)). This is basically where I am stuck.

    Thank you
  5. Sep 13, 2010 #4
    ln(ln(x)) is correct.
    Regarding Integration by parts, you have missed -ve sign in differentiating u
    It will be
    resulting in 1=0 which is wrong.
  6. Sep 13, 2010 #5


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    Gold Member

    No, it results in 1 = C.
  7. Sep 13, 2010 #6
    I can see I've missed the negative which changes it quite a bit.

    Is the integration by parts correct for C=1? It hasn't really solved the integral, or am I missing something here?
  8. Sep 13, 2010 #7


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    Gold Member

    Typically, you'll see this problem as an example of why it's so important to remember the constant of integration, because otherwise you end up with nonsense like 1 = 0.
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