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Integration By Substitution

  1. Jun 12, 2010 #1
    I'm attempting to solve the following problem:

    [tex]\int_{0}^{\infty} {\frac{x arctan(x)}{(1+x^{2})^{2}}dx}[/tex]

    I started with a substitution:

    [tex]u=arctan(x), du=\frac{1}{(1+x^{2})}dx[/tex]

    This seemed like the right thing to do, but after trying to put it together in several different ways I got nowhere. I looked at what WolframAlpha had to say. It got this after doing the same substitution:

    [tex]\int_{}^{} {u sin(u)cos(u)du}[/tex]

    I've gone at this for over half an hour now and I've gotten nowhere. Some insight into how this step was made would be appreciated.
  2. jcsd
  3. Jun 12, 2010 #2
    What will happen if you make u=(1+x^2)^2?
  4. Jun 13, 2010 #3
    I got this:

    [tex]\int_{}^{} {\frac{arctan(x)}{u*4\sqrt{u}du}[/tex]
    which can be:
    [tex]\int_{}^{} {\frac{arctan(x)}{4u^{\frac{3}{2}}}du}[/tex]

    It's not incredibly helpful. I see no way of dealing with that arctangent.
  5. Jun 13, 2010 #4
    If u = arctan(x), what does x equal?
  6. Jun 13, 2010 #5


    Staff: Mentor

    There is still a factor of x in the numerator that is unaccounted for, so this substitution isn't feasible.
    Use integration by parts, with u = arctan(x) and dv = xdx/((1 + x^2)^2). The resulting integral can be evaluated using a trig substitution.

    The original integral is improper because of the infinity as one of the limits of integration, so you will need to take a limit at some point. One way to go about this is to evaluate this integral:
    [tex]\int_0^b \frac{x~arctan(x)~dx}{(1 + x^2)^2}[/tex]

    Your result of this integral will involve b, so take the limit as b goes to infinity to get your final answer.
  7. Jun 13, 2010 #6
    I'll try that solution out. I usually solve the indefinite integral first, then work out the proper solution for the definite integral with the limit.
    [tex]-\frac{arctan(x)}{2(1+x^{2})}+\frac{1}{2}\int_{}^{} {(\frac{1}{(1+x^{2})})^{2}dx}[/tex]

    I don't see a trig substitution working for that integral. I'll keep working at it.
    Last edited: Jun 13, 2010
  8. Jun 14, 2010 #7


    User Avatar
    Homework Helper


    Substitute x = tanθ, the dx = sec^2(θ)dθ

    Integration becomes


    Substitute [tex]cos^2(\theta)= \frac{1}{2}(1 + cos2(\theta))[/tex]

    Now solve the integration.
    Last edited: Jun 14, 2010
  9. Jun 15, 2010 #8
    Yup, I managed to get it. Thanks for the help.
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