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Math Help - Improper integrals

  1. #1
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    Improper integrals

    Have been going through evaluating improper integrals by expressing them as limits and have got to the following,

    integral from 0 to infintiy of 1/(x^3) with respect to x,

    evaluating this from 1 to infinity would seem to make sense to me but from 0 i'm unsure how to proceed.

    Thanks for any help in advance
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  2. #2
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    Quote Originally Posted by monster View Post
    Have been going through evaluating improper integrals by expressing them as limits and have got to the following,

    integral from 0 to infintiy of 1/(x^3) with respect to x,

    evaluating this from 1 to infinity would seem to make sense to me but from 0 i'm unsure how to proceed.

    Thanks for any help in advance
    \int_0^{\infty}{\frac{1}{x^3}\,dx} = \lim_{a \to 0}\lim_{b \to \infty}\int_a^b{\frac{1}{x^3}\,dx}

     = \lim_{a \to 0}\lim_{b \to \infty}\left[-\frac{1}{x^4}\right]_a^b

     = \lim_{a \to 0}\lim_{b \to \infty}\left[\frac{1}{a^4} - \frac{1}{b^4}\right]

    = \lim_{a \to 0}\left[\frac{1}{a^4} - 0\right]

     = \lim_{a \to 0}\frac{1}{a^4}

     = \infty.


    The integral is divergent.
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