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Math Help - Integrals problem

  1. #1
    MHF Contributor Also sprach Zarathustra's Avatar
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    Integrals problem

    Let k, s be natural numbers, prove that the function :

    f(x) = x^(-k) * sin(x^s) for x in (0,1] and f(x) = 0 for x=0

    is integrable in [0,1] if and only if k<=s.


    Thank you!
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  2. #2
    Member Black's Avatar
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    Note that f is continuous and positive on (0,1).

    If s \ge k, then we have

    \int_0^1\frac{\sin(x^s)}{x^k}dx < \int_0^1\frac{x^s}{x^k} \, dx = \int_0^1 x^{s-k}dx < \infty .

    If s < k, note that

    \lim_{x \to 0^{+}}\frac{\sin(x^s)}{x^k}=\infty.
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