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(Improper Integrals) Not sure if I'm doing this properly

  1. Jan 14, 2013 #1
    Initial improper integral:
    ∫ dx / (1+x**2) * (1+ atan(x)) , x = 0, ∞

    Substitutions:
    μ = 1 + atan(x)
    dμ = dx / (1 + x**2)

    μ(∞) = 1 + pi/2
    μ(0) = 1

    Integral:
    ∫ dμ / μ , μ = 1, 1+ pi/2

    Then solve.

    I'm getting the right answer, but I think I'm botching something due to a lack of understanding about how notation works with indefinite integrals. Is this right?
     
  2. jcsd
  3. Jan 14, 2013 #2

    Mark44

    Staff: Mentor

    This is an improper integral, so you're going to need to evaluate a limit.

    $$ \int_0^{\infty} \frac{(1 + arctan(x))dx}{1 + x^2}$$
    $$ = \lim_{b \to \infty} \int_0^b \frac{(1 + arctan(x))dx}{1 + x^2}$$

    The integral itself is fairly easy - you have the right substitution.
     
  4. Jan 14, 2013 #3

    Zondrina

    User Avatar
    Homework Helper

    It is indeed an improper integral, and the fact you can find a finite value for it implies that the integral converges.
     
  5. Jan 14, 2013 #4

    Ray Vickson

    User Avatar
    Science Advisor
    Homework Helper

    What you wrote means
    [tex] \int_0^{\infty} \frac{1 + \arctan(x)}{1+x^2} \, dx.[/tex] Is that what you meant, or did you mean
    [tex] \int_0^{\infty} \frac{1}{(1 + x^2)(1 + \arctan(x)}\, dx ?[/tex]
     
  6. Jan 14, 2013 #5

    Mark44

    Staff: Mentor

    Based on context, I would say that the OP meant the first one, above.
     
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