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: Finding convergence/divergence of improper integral

  1. Mar 18, 2012 #1
    Determine the convergence or divergence of this following improper integral:

    [tex]\int_2^∞ \frac {1}{(x^3+7)^{\frac{1}{3}}}[/tex]

    So I'm trying to find something easy to compare this to, any help?
  2. jcsd
  3. Mar 18, 2012 #2
    Well, as I'm sure you can tell, it is a lot like 1/x. So, I would try to find something that is close to 1/x to which you can compare this.
  4. Mar 18, 2012 #3
    Why can't I just compare it to 1/x?
  5. Mar 18, 2012 #4
    I don't know if you have gone over this, but you can compare it to

    [tex]\int\limits_1^\infty {\frac{1}{{{x^n}}}} [/tex]

    If n<2, it is divergent. If n≥2, then it is convergent.
    Last edited: Mar 18, 2012
  6. Mar 18, 2012 #5
    [tex]\frac{1}{(x^3+7)^{\frac{1}{3}}} < \frac{1}{x} [/tex]

    isn't it? Since [
    [tex]\int_2^∞ \frac {1}{x}[/tex]
    diverges, this isn't any help.

    So, I'd try showing that
    [tex](x^3+7)^{1/3} < x + b[/tex]

    where b is a number. (In particular, b is a number that appears in your problem).

    Then realize that

    [tex]\int_2^∞ \frac {1}{x+b}[/tex]

    diverges, and you're done.
  7. Mar 19, 2012 #6
    How can I show that [tex]\int_2^∞ \frac {1}{x+7}[/tex] is divergent?
  8. Mar 19, 2012 #7
    First of all, I was kind of wrong when I said that "b" was in your problem. Use [tex]7^{1/3}[/tex] in stead of [tex]7[/tex]. As for showing that

    [tex]\int_2^∞ \frac {1}{x+7^{1/3}}[/tex]

    is divergent, is there some theorem in your book that would help? I don't know if there is or not. But, you could say that

    [tex]lim \frac{1}{x+7^{1/3}} = \frac{1}{x}[/tex]

    and argue from there. Or, you could just integrate

    [tex]\int_2^∞ \frac {1}{x+7^{1/3}}[/tex]

    You have done so to show that

    [tex]\int_2^∞ \frac {1}{x}[/tex]

    is divergent, right? Just use a change of variables.
Share this great discussion with others via Reddit, Google+, Twitter, or Facebook