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Homework Help: Improper Integrals

  1. Jan 26, 2007 #1
    1. The problem statement, all variables and given/known data

    Evaluate the following improper integrals of explain why they don't converge.
    Integral from 0 to infinity(1/the cubed root of x)dx
    I'm not sure how to make forulas, so this is the best I can do:
    0∫∞ (1/(3∙√x))dx

    2. Relevant equations

    No equations

    3. The attempt at a solution

    I know that when there is ∞ as an upper bound, the intergration is changed to:

    lim as b→∞ 0∫b (1/(3∙√x))dx
    But in this form, the 0 is a problem.

    and if the lower bound, 0, causes the function to be undefined, the integration is changed to:

    lim as a→0+ a∫∞ (1/(3∙√x))dx
    But, in this for the infinity is still a problem.

    Is there any way to combine the two so I can solve this.
    Any help is appreciated.
  2. jcsd
  3. Jan 26, 2007 #2


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    Well, the first thing you had better do is actually write out the anti- derivative!
    What is [tex]\int \frac{1}{^3\sqrt{x}}dx= \int x^{-\frac{1}{3}}dx[/tex]?

    Does it converge as x goes to 0? What happens as x goes to infinity?

    Oh, and notice that the problem specifically asks you to "explain why they don't converge". Maybe the problem you are having isn't really a problem!
    Last edited by a moderator: Jan 26, 2007
  4. Jan 26, 2007 #3


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    Is there anything wrong with the most obvious approach: make both changes?
  5. Jan 26, 2007 #4
    The anti- derivative is X^(2/3)
    As x goes to infinity, the anti-derivative goes to infinity.
    As x goes to 0, the anti- derivative goes to 0.

    so, would I evaluate it as (infinity - 0), which is infinity, therefore it diverges.

    Is this right?
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