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Improper Integral

  1. Feb 23, 2008 #1
    1. The problem statement, all variables and given/known data
    Does the integral from 0 to Infinity of [tex]\int\frac{1}{\sqrt x \sqrt{x+1}\sqrt{x+2}}dx[/tex] converge or diverge?


    2. Relevant equations
    None.


    3. The attempt at a solution

    I tried to integrate it, but I haven't even been able to do that so I couldn't then evaluate the limit of the integral.
     
    Last edited: Feb 23, 2008
  2. jcsd
  3. Feb 23, 2008 #2

    arildno

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    Well, since you can't use an anti-derivative of the integrand to answer the question, can you use some other technique?

    To give you a hint:
    Were you asked to actually CALCULATE the value of the integral, or just whether or not the integral converged or diverged?
     
  4. Feb 23, 2008 #3
    Wow, I'm getting dizzy looking at your problem. Is this correct ...

    [tex]\int\frac{1}{\sqrt x \sqrt{x+1}\sqrt{x+2}}dx[/tex]
     
  5. Feb 23, 2008 #4
    Yea that is correct, and the question does just ask about its convergence.
     
  6. Feb 23, 2008 #5

    arildno

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    Now:

    Note that this integral is improper in two radically different ways:
    1. At x=0 (lower limit), the integrand is infinitely large.
    2. The upper limit is not a real number.

    Now, let us first try to analyze two distinct cases, where each of these has only one of these pathological traits:

    a) Suppose that we investigate the case where only 1 holds, say that the integral goes from x=0 to x=1.
    Does this integral converge or diverge?

    b) Suppose we investigate the other special case, say looking at the integral from say..x=1 to infinity. Does this integral converge or diverge?


    What can you conclude from your analysis?
     
    Last edited: Feb 23, 2008
  7. Feb 23, 2008 #6
    I am fairly that both of those cases converge, although I had to use a Ti-89 to evaluate them. Since they both converge, the entire integral should converge as well. Though, I am struggling to reach the convergence of each case without an 89.
     
  8. Feb 23, 2008 #7

    arildno

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    Well, can you compare each case with two simpler integrals that you CAN calculate, and utilize those results to reach your (correct) conclusion?

    In particular, in terms of inequalities, what must be your guiding principle to determine whether your original (part-)integrals must converge as well?
     
  9. Feb 23, 2008 #8
    Use integral test for series; hopefully I am not making forced connection between the two things.

    So, your integral would converge if the series converges

    nvm.. this does not work

    or.. does works ...
    1/n^1.5 is similar to your series which converges
     
    Last edited: Feb 23, 2008
  10. Feb 23, 2008 #9
    I am stuck with which functions that are larger for all x, that will converge. Help?
     
  11. Feb 24, 2008 #10

    arildno

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    A good idea!

    Now, instead of finding a single, continuous function that is larger than the integrand for ALL non-negative x, try first to find a nice function that is larger than the integrand for x between 0 and 1, and whose integral you know will converge.
     
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