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Math Help - WHY is this integral divergent? (another question)

  1. #1
    s3a
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    WHY is this integral divergent? (another question)

    Why is this integral divergent?:


    I get 1/2 * (ln 1 - ln(inf) + ln(inf) - ln 1) so I thought the answer would be zero. Or do we assume that the infinities are different in each ln therefore it diverges? Or am I completely off?

    Any input would be greatly appreciated!
    Thanks in advance!
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  2. #2
    Member Black's Avatar
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    Split the integral up

    \int_{-\infty}^{\infty}\frac{x}{x^2+11} \, dx=\int_{-\infty}^{0}\frac{x}{x^2+11} \,dx+\int_{0}^{\infty}\frac{x}{x^2+11} \, dx.

    Let's do the second integral on the right.

    \int_{0}^{\infty}\frac{x}{x^2+11} \, dx= \frac{1}{2} \lim_{t \to \infty} \left[ \ln(x^2+11)\right]_{0}^t= \infty.

    Therefore, the integral \int_{-\infty}^{\infty}\frac{x}{x^2+11} \, dx is divergent.
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  3. #3
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    [/tex]lim_{a\to\infty}\int_{-a}^a f(x)dx[/tex] is often called the "Cauchy Principal Value" of \int_{-\infty}^\infty f(x) dx and often exists when the integral itself, which is defined as \lim_{a\to -\infty}\lim_{b\to\infty} \int_a^b f(x) dx does not exist.

    Here, the integral intself, where the two limits must be taken independently does not exist- what you calculated is the "Cauchy Principal Value".
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