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Find the values of p for which the integral converges?

  1. Jan 21, 2012 #1
    1. The problem statement, all variables and given/known data
    Find the values of p for which the integral converges and evaluate the integral for those values of p.

    ∫ 0->1 1/(x^p) dx

    2. Relevant equations
    None.


    3. The attempt at a solution

    First thought:

    Since we must evaluate 0 to 1, 1/0 is undefined so maybe 1/ (0^0) = 1.
    I don't think this is correct.

    Second thought:
    If the first thought didnt work, how about lim x->0.
    Test:
    p≥1
    1/ (0.000000000000000000...01)^1
    ≈ ∞

    0>p>1
    1/ (0.000000000000000000...01)^.5
    ≈ ∞

    p<0
    1/ (0.000000000000000000...01)^-1
    ≈ 0

    If this test is true, I am not sure how to evaluate for 1.

    4. Answer in the back of the testbook
    p<1 , 1/(1-p)

    From the answer, I am not sure how p<1 would work (but I do understand p<0).
     
  2. jcsd
  3. Jan 21, 2012 #2

    Dick

    User Avatar
    Science Advisor
    Homework Helper

    You are making a test to guess if f(x)=1/x^p diverges at x=0. That doesn't tell you the integral necessarily diverges. Take your p=1/2 case. What the antiderivative of 1/x^(1/2)?
     
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