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Improper Integral

  1. Oct 23, 2015 #1
    1. The problem statement, all variables and given/known data
    ∫(sin(x)+2)/x^2 from 2 to infinity. Determine if this improper integral converge or diverge.


    2. The attempt at a solution
    lim(x→infinity)=∫(sin(x)+2)/x^2 from 2 to t.

    I know that if the integral ends up to be an infinite number, this will be converge otherwise, it will be diverge. However, I couldn't find a way to integrate this function. When I tried to graph it, I see that as x approaches to infinity, the function is getting closer to zero and not equal to. Is that means that it is diverge since even though the function is getting smaller and smaller, the area is still increasing.
     
  2. jcsd
  3. Oct 23, 2015 #2

    Mark44

    Staff: Mentor

    ## -1 + 2 \le \sin(x) + 2 \le 1 + 2##
    Does that help?
     
  4. Oct 23, 2015 #3
    Can I do (−1+2)/x^2 ≤ sin(x)/x^2+2 ≤(1+2)/x^2, then integrate 1/x^2 and 3/x^2 from 2 to ∞?
    Then the answer will be 1/2 ≤ ∫(sin(x)+2)/x^2 ≤ 3/2, therefore, it will be converge?
     
  5. Oct 23, 2015 #4

    Mark44

    Staff: Mentor

    Yes, that's the idea.
     
  6. Oct 23, 2015 #5
    Thank you for helping!
     
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