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: Convergence of integral

  1. Apr 11, 2009 #1
    i am given the following, and asked if it converges or diverges

    [tex]\int[/tex]dx/ln(x) (from 0-1)

    what i did was look for a similar function
    i took 1/x which is bigger than 1/lnx,
    but 1/x diverges from 0-1

    can i split my function up and compare
    [tex]\int[/tex]dx/ln(x) (from 0-0.5) + [tex]\int[/tex]dx/ln(x) (from 0.5-1)
    now i know that 1/x converges from 0.5-1 so 1/lnx does as well

    now i need to find a fuction to compare 1/lnx with from 0-0.5
    1/lnx < 1/(x+1)
    1/(x+1) converges
    therefore 1/lnx does as well

    therefore 1/lnx is made up of two converging parts from 0-1 so the whole also converges

    is this correct???
  2. jcsd
  3. Apr 11, 2009 #2
    The integral is finite for any interval (k,1), where 0 < k < 1, because continuous, bounded functions are always integrable. You can't compare 1/ln(x) to 1/x or 1/(x+1) for that matter because the technique you're working with only works for positive functions. We're trying to check that the integral isn't NEGATIVE infinity, so finding a function that's larger than it on an interval doesn't work. You need to find a function that decreases to -infinity as x goes to 0 SLOWER than 1/ln(x) on an arbitrary small interval around 0. Considering that ln(x) goes to -infinity as x goes to 0 at the same rate that e^x goes to infinity as x goes to infinity, what could you use?
  4. Apr 11, 2009 #3
    i dont know, 1/e^x ??
    i understand what you are saying but dont understand how to put it into practice
  5. Apr 12, 2009 #4
    i tried 1/(e^1/x) but i found that this converges and is smaller than 1/lnx so it doesnt help, please give some idea of what to do
  6. Apr 12, 2009 #5
    use the integral test, by using integration by parts.
  7. Apr 12, 2009 #6
    how do i integrate in parts, i tried and i keep on going in cirlces!

    i made t=lnx, dt=dx/x

    then got

    integral of ((e^t)/t)dt, made 1/t=u e^tdt=dv, and even tried the other way round, -nothing!!
Share this great discussion with others via Reddit, Google+, Twitter, or Facebook