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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!!
     
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