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Math Help - improper integral

  1. #1
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    improper integral

    Show that the improper integral of (x^(-p))(cos x)dx from 0 to 1 converges iff p < 1.
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  2. #2
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    Quote Originally Posted by allrighty View Post
    Show that the improper integral of (x^(-p))(cos x)dx from 0 to 1 converges iff p < 1.
    I assume that 0 < p in order to make it improper.

    Note that for 0 \leq x \leq 1 we have 0\leq \cos x \leq 1 - \frac{x^2}{2!}+\frac{x^4}{4!}.

    This means, 0\leq x^{-p} \cos x \leq x^{-p} \implies 0 \leq \int_0^1 x^{-p} \cos x ~ dx \leq \int_0^1 x^{-p} ~ dx = \frac{1}{1-p}.

    Why does it feel as it I did something wrong?
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