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Math Help - Condition of an exponent for convergence of an integral

  1. #1
    MHF Contributor arbolis's Avatar
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    Condition of an exponent for convergence of an integral

    I must find for which values of p \in \mathbb{R} the following integral converges : \int_e^{+\infty} \frac{dx}{x\ln^p|x|}. Mathstud28 already done that (well, something pretty similar) in one of my earlier threads, but I didn't understand a step he did, so I would like a little bit more explanations :
    [quote]How about when
    Hmm, sorry the quote doesn't work. I copy and past it, but it only shows the first line and all the other is cut, so I quote it by parts :[quote].
    Still doesn't work!
    Anyway, he was working with \int_2^{+\infty}\frac{dx}{x\ln^p|x|}. He said that if 0<p<1, then the indefinite integral is equal to \frac{\ln^{-p+1}(x)}{-p+1}. This is what I missed to understand. Did he do integration by parts? How could he reach this result? Thanks!!
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  2. #2
    MHF Contributor Mathstud28's Avatar
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    Still doesn't work!
    Anyway, he was working with \int_2^{+\infty}\frac{dx}{x\ln^p|x|}. He said that if 0<p<1, then the indefinite integral is equal to \frac{\ln^{-p+1}(x)}{-p+1}. This is what I missed to understand. Did he do integration by parts? How could he reach this result? Thanks!!
    If p\ne{-1}

    Then \int\frac{dx}{x\ln^p(x)}=\int\frac{\frac{dx}{x}}{\  ln^p(x)}

    Now let u=\ln(x)
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  3. #3
    Eater of Worlds
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    This is mathstud's baby, so I will leave him answer. I am sure he is typing as I write this. Anyway, here is a site with a list of integrals you may find helpful instead of deriving them each time. Unless you want to.

    Definite Integrals, General Formulas Involving Definite Integrals
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  4. #4
    MHF Contributor arbolis's Avatar
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    Thanks mathstud28, I reached it! I thought it would have been much more complicated.
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  5. #5
    MHF Contributor Mathstud28's Avatar
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    Quote Originally Posted by galactus View Post
    I am sure he is typing as I write this.
    Or finishing before you
    Quote Originally Posted by arbolis View Post
    Thanks mathstud28, I reached it! I thought it would have been much more complicated.
    Yeah, sometimes the obvious alludes even the best of us.
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  6. #6
    Eater of Worlds
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    Hey mathstud, you know what might be fun?. Go to the link I posted above and actually derive the solutions they give for the integrals. You have probably done that already.

    Some of them are probably very challenging.
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