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Homework Help: Find Integral Convergence

  1. Jun 22, 2010 #1
    Hello,
    I need to find the values of alpha for which the following integral converges:
    [tex]\int x^\alpha*ln(x)[/tex] the integral is from 0 to 1.

    I don't really know which test should I use or how to calculate the limit of the integral as x->o+
     
  2. jcsd
  3. Jun 22, 2010 #2
    Please see attachment for some insight.
     

    Attached Files:

  4. Jun 22, 2010 #3
    I still can't see how to find the alphas given that statement after you integrated
     
  5. Jun 22, 2010 #4
    This might not be the correct approach to the problem, I was just trying to give you some insight that perhaps would be helpful. Hopefully someone will clarify this further.
     
  6. Jun 23, 2010 #5
    I'm sorry for the bump. Can anyone please help?
     
  7. Jun 23, 2010 #6
    It's really all in proceeding from the hint given in the attachment.

    You end up with

    [tex] \left(\frac{1}{\alpha+1}x^{\alpha+1}\ln(x)\right)\bigg|_0^1 - \left(\frac{1}{(\alpha+1)^2}x^{\alpha+1}\right)\bigg|_0^1[/tex]

    This will exist if (and only if) the two limits

    [tex] \lim_{x\to 0^+}\frac{1}{\alpha+1}x^{\alpha+1}\ln(x),\quad
    \lim_{x\to 0^+}\frac{1}{(\alpha+1)^2}x^{\alpha+1}[/tex]

    both exist. Use L'Hospital's rule for the first (write it with [tex] x^{-\alpha-1}[/tex]) on bottom, and the second is immediate. You should get the same answer for both
    limits working out.
     
  8. Jun 24, 2010 #7
    If you have general integral

    [tex]\int_{a}^{b} f(x) dx[/tex] and want to find the limit then here is where you do

    Assuming that f is continuous on (a,b) and not continuous at x = a.

    [tex]\int_{a}^{b} f(x) dx = \lim_{t \to a^+} \int_{t}^{b} f(x) dx = \lim_{t \to a^+} F(x)|_{t}^{b}[/tex]
     
    Last edited: Jun 24, 2010
  9. Jun 24, 2010 #8
    I understand this steps and everything but I still can't see how to get a specific alphas for which the integral converges. How can I find them?
     
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