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Generalize improper integral help

  1. Feb 26, 2015 #1
    1. The problem statement, all variables and given/known data
    Generalize the integral from 0 to 1 of 1/(x^p)
    What conditions are necessary on P to make the improper integral converge and not diverge?

    I believe I have the answer but I would like to make it more formal and sound. Can someone help me with that?


    2. Relevant equations
    None

    3. The attempt at a solution
    Lim(b--> infinity) of ∫ 1/(x^p)dx (from 1 to b)
    = lim(b--> ∞) of ∫ x^(-p)dx (from 1 to b)
    = lim(b-->∞) of (x^(-p +1))/(-p+1) evaluated (from 1 to b)
    = lim(b--> ∞) of [(b^(-p + 1))/(-p+1) - (1^(-p+1))/(-p+1))]

    From here I would normally apply the limit to the last thing listed but it seems there are some constraints immediately needed.

    P ≠ 1 also x ≠ 0
    how would I figure out what constraints need to be put on P for the improper integral to converge?

    By intuition I think I figured out the solution. I would need the [(b^(-p + 1))/(-p+1) portions numerators power to go negative so we can have it tend to 0 rather than infinity.

    Therefore I conclude that P>1 for the integral to converge.
     
  2. jcsd
  3. Feb 26, 2015 #2

    Dick

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    Science Advisor
    Homework Helper

    The integral of 1/x^2 from 0 to 1 does not converge even though 2>1. I'm really not sure why you are taking the integral from 1 to b and letting b->infinity if you want the integral from 0 to 1.
     
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