1. Limited time only! Sign up for a free 30min personal tutor trial with Chegg Tutors
    Dismiss Notice
Dismiss Notice
Join Physics Forums Today!
The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

Homework Help: Integration by Parts using Ln(x)

  1. Jun 8, 2013 #1
    1. The problem statement, all variables and given/known data

    [tex]\int_4^5 \frac{dx}{(3x)(ln(x))(ln^3(ln(x)))} \,[/tex]

    2. Relevant equations

    I'll probably need to use u-substitution as well as integration by parts for this problem.

    3. The attempt at a solution
    [tex]\int_4^5 \frac{dx}{(3x)(ln(x))(ln^3(ln(x)))} \,[/tex]

    1. Factor out 1/3 for convenience.

    [tex]\frac{1}{3}\, \int_4^5 \frac{dx}{(x)(ln(x))(ln^3(ln(x)))} \,[/tex]

    2. Let u = ln(x), du = 1/x * dx, thus dx = du * x

    [tex]\frac{1}{3}\, \int_4^5 \frac{(x)(du)}{(x)(u)(ln^3(u))} \, = \frac{1}{3}\, \int_4^5 \frac{du}{(u)(ln^3(u))} \,[/tex]

    From integration by parts we know that ∫g*dv = g*v - ∫dg*v

    I'm stuck here with regards to what to assign as g and what to assign as dv.

    I could make:
    g = 1/u
    dg = -1/u^2

    dv = 1/ln^3(u) * du
    v = ????

    I'm not sure how to integrate dv to get to v to make the integration by parts work.

    Any help would be appreciated.
  2. jcsd
  3. Jun 8, 2013 #2
    Make a second substitution
  4. Jun 8, 2013 #3


    User Avatar
    Homework Helper

    Integration by parts is not helpful here

    You wither want two use the substitution

    u=(log log x)^3

    or equivalently another substitution after yours

    u=log x
    w=(log u)^3
  5. Jun 8, 2013 #4
    That second substitution complicates the problem, a simple w=lnu is enough
  6. Jun 8, 2013 #5
    Thanks for your help, guys. I think I understand what you're saying.

    Let g = ln(u), dg = 1/u * du, thus du = dg* u

    Substituting back in I get:

    [tex]\frac{1}{3}\, \int_4^5 \frac{dg*u}{(u)(g^3)} \, = \frac{1}{3}\, \int_4^5 \frac{dg}{g^3} \, = -\frac{1}{2g^2}\,\frac{1}{3}\, [/tex]

    and then I would substitute back to reobtain x and apply the limit from 4 to 5.

    Thank you, guys. Much appreciated.
Share this great discussion with others via Reddit, Google+, Twitter, or Facebook

Have something to add?
Draft saved Draft deleted