Integrating 1/xln(x) using integration by parts

  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. It is correct
  4. 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. 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. jgens

    jgens 1,621
    Gold Member

    No, it results in 1 = C.
  7. 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. jgens

    jgens 1,621
    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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