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: Improper integral comparison test

  1. Jan 3, 2015 #1
    • Member warned about not using the homework template
    The question asks whether the following converges or diverges.

    [tex]\int_{0}^{\infty } \frac{\left | sinx \right |}{x^2} dx[/tex]

    Now I think there might be a trick with the domain of sine function but I couldn't make up my mind on this.
    I tried to compare it with 1/x^2, (sinx)/x, and sinx. I actually expected that I would get something good with 1/x^2, but as the lower limit of the integral is zero, it ended up with infinity on (0, inf) and since 1/x^2 is greater than (sinx)/x^2, and is divergent as we just found, we cannot say whether the given function diverges or converges. So I'm wondering what is the right track on this question?
    Last edited: Jan 3, 2015
  2. jcsd
  3. Jan 3, 2015 #2

    Stephen Tashi

    User Avatar
    Science Advisor

    [tex]\int_{0}^{\infty } \frac{\left | sinx \right |}{x^2} dx = [/tex]

    [tex] \int_{0}^{ \frac{\pi}{2} } \frac{\left | sinx \right |}{x^2} dx + \int_{ \frac{\pi}{2} }^{\infty} \frac{\left | sinx \right |}{x^2} dx [/tex]

    Deal with the two integrals separately.

    In the interval [itex] [0,\frac{\pi}{2}] [/itex] you can establish an inequality between [itex] |\sin(x)| [/itex] and a non-periodic function. Try comparing it to [itex] f(x) = x [/itex].
Share this great discussion with others via Reddit, Google+, Twitter, or Facebook

Have something to add?
Draft saved Draft deleted