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Math Help - MTH101 Q1_b

  1. #1
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    MTH101 Q1_b

    show that the given integral converges or not
    ∫1/xln^3x dx from e to +∞
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  2. #2
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    Quote Originally Posted by TAHIR View Post
    show that the given integral converges or not
    ∫1/xln^3x dx from e to +∞
    \int \frac{1}{x\ln ^3 x} dx

    Let t=\ln x \implies t' = 1/x

    \int \frac{1}{t^3} dt = -\frac{1}{t^2}+C = -\frac{1}{\ln^2 x} +C

    Thus,
    \int_e^{\infty} \frac{1}{x\ln^3 x} = \lim_{N\to \infty} - \frac{1}{\ln ^2 N} + e = e

    It converges.
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  3. #3
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    Quote Originally Posted by ThePerfectHacker View Post
    \int \frac{1}{x\ln ^3 x} dx

    Let t=\ln x \implies t' = 1/x

    \int \frac{1}{t^3} dt = -\frac{1}{t^2}+C = -\frac{1}{\ln^2 x} +C

    Thus,
    \int_e^{\infty} \frac{1}{x\ln^3 x} = \lim_{N\to \infty} - \frac{1}{\ln ^2 N} + e = e

    It converges.
    Is your result correct? My answer is 1/2.
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