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Improper Integral Question

  1. Feb 4, 2012 #1
    EDIT: I am sorry if you can't understand the integral straight away, I am not familiar with using the notion provided by this forum. I tried, but...

    1. The problem statement, all variables and given/known data
    Find all values of the constant α for which the integral:
    ∫ [(x/(x^2 + 1)) - (3a/(3x + 1))] dx (from 0 to +infinitity)
    converges. Evaluate the integral for these values of α (as a function of α).

    3. The attempt at a solution
    I don't really know how to approach this. It seems simpler to separate it into 2 separate integrals. And in the integral with the constant 'a', I can sub u = 3x + 1, then du = 3dx
    Making it: adu/x

    I don't really know if I am on the right track.

    Thanks for reading this!
    Last edited: Feb 4, 2012
  2. jcsd
  3. Feb 4, 2012 #2


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    You probably mean adu/u for your substitution. You are started on the right track. Work out both integrals and get the antiderivative as a function of x. Once you get that start working on what the limit of it is as x->infinity. Using the rules of logs will help.
  4. Feb 4, 2012 #3
    Yes, adu/u. My fault. And thanks!
  5. Feb 7, 2012 #4
    OK, now I am confused. The ln|3x+1| as x goes to infinity is infinity. So it diverges. But how do I make it converge? Same with the first part of the integral...
  6. Feb 7, 2012 #5


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    Combine the two integral expressions. Use rules of logs. For a special value of a the divergences might cancel.
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