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Problem with divergent integral

  1. Oct 29, 2009 #1
    I'm confused with the following integral.

    Let a > 1.

    [tex] \int_{a}^{\infty} \left( \dfrac{1}{t} - \dfrac{1}{t-1} \right) ~ dt = \left[ \log \left| \dfrac{t}{t-1} \right| \right]_{a}^{\infty} = - \log \left| \dfrac{a}{a-1} \right| [/tex]

    This should be the correct result. But I could also decompose the integral into two parts (because the integrand is a sum) and compute:

    [tex] \int_{a}^{\infty} \left( \dfrac{1}{t} - \dfrac{1}{t-1} \right) ~ dt = \left| \log \vert t \vert \right|_{a}^{\infty} - \left[ \log \vert t - 1 \vert \right]_{a}^{\infty} = - \log \left| \dfrac{a}{a-1} \right| + \infty - \infty [/tex]

    But [tex] \infty - \infty [/tex] is of course not defined!

    Where did I make a mistake? I don't find it.
     
  2. jcsd
  3. Oct 29, 2009 #2
    Decomposing the integral into two parts is justified only if both integrals are finite.
     
  4. Oct 29, 2009 #3
    Thanks :smile:
     
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