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: Find Integral Convergence

  1. Jun 22, 2010 #1
    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?
Share this great discussion with others via Reddit, Google+, Twitter, or Facebook