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: Indefinite integral and proving convergence

  1. Oct 2, 2015 #1
    1. The problem statement, all variables and given/known data

    okay so the equation goes:

    ∫(x*sin2(x))/(x3-1) over the terminals:
    b= ∞ and a = 2

    2. Relevant equations

    Various rules applying to the convergence or divergence of integrals such as the p-test, ratio test, squeeze test etc

    3. The attempt at a solution

    Okay so I have tried factorising the bottom, I have tried the squeeze test and the ratio test to no success..

    would method should I be applying, because I am unsure how to split up this integral...

  2. jcsd
  3. Oct 3, 2015 #2
    Here's a hint: first notice that [itex]\sin^2 x \leq 1[/itex] for all [itex]x[/itex]. Then you can compare your integral with another one whose integrand is always greater. You can then prove the convergence of that integral.
  4. Oct 3, 2015 #3
    Therefore using the comparison test should i compare this integral to 1/x3 over the same interval?
  5. Oct 3, 2015 #4


    User Avatar
    Homework Helper

    Lets apply a useful theorem. If ##\int_a^b |f(x)| \space dx## converges, then ##\int_a^b f(x) \space dx## converges. Note the converse of this theorem is not true. So you should always determine the convergence of ##\int_a^b |f(x)| \space dx## to determine if ##\int_a^b f(x) \space dx## converges.

    Applying this theorem to your given problem:

    $$\int_{2}^{\infty} \left| \frac{x \sin^2(x)}{x^3 - 1} \right| \space dx = \int_{2}^{\infty} \frac{|x| |\sin^2(x)|}{|x^3 - 1|} \space dx \leq \int_{2}^{\infty} \frac{|x|}{|x - 1| |x^2 + x + 1|} \space dx$$

    For ##x \in [2, \infty)##, we can remove the absolute values in the expression because all of the terms would be positive anyway. So the new question is, does this integral converge:

    $$\int_{2}^{\infty} \frac{x}{(x - 1) (x^2 + x + 1)} \space dx$$

    Hint: Start with a partial fraction expansion, and then complete the square.
  6. Oct 3, 2015 #5
    Right now I understand. So the absolute of f(x) is always said to converge but the the opposite isn't always true ?

    Right do we are performing a comparison test to x/(x^3-1)
Share this great discussion with others via Reddit, Google+, Twitter, or Facebook

Have something to add?
Draft saved Draft deleted