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Math Help - Integral-serie

  1. #1
    Super Member dhiab's Avatar
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    Integral-serie

    Calculate and student this integral :
    \int_0^{ + \infty } {\left( {\frac{{\arctan \left( {\frac{1}{x}} \right)}}{{x^2 + 1}}} \right)} dx<br />
    Same question :
    \int_0^{ + \infty } {\left( {\sin \left( {\frac{1}{{x^\alpha }}} \right)} \right)} dx.....\left( {\alpha \in \Re } \right)
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  2. #2
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    Quote Originally Posted by dhiab View Post
    Calculate and student this integral :
    \int_0^{ + \infty } {\left( {\frac{{\arctan \left( {\frac{1}{x}} \right)}}{{x^2 + 1}}} \right)} dx<br />
    Same question :
    \int_0^{ + \infty } {\left( {\sin \left( {\frac{1}{{x^\alpha }}} \right)} \right)} dx.....\left( {\alpha \in \Re } \right)
    You haven't shown any working, you rarely do. I can never tell if you're asking for help or posing a question which you know the answer to.

    If you are seeking help then you should really show what working you have done and where you get stuck.

    Here's a hint for the first one

    \frac{d}{dx} \left(\arctan (1/x) \right) = -\frac{1}{1+x^2}

    Hope this helps.
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  3. #3
    Math Engineering Student
    Krizalid's Avatar
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    As for the second question, the integral obviously diverges for \alpha\le0, so the only interesting case to analyze convergence is \alpha>0. Here's a Hint: the integral converges for all \alpha>1. Prove it.
    Last edited by Krizalid; January 2nd 2010 at 02:34 PM.
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  4. #4
    MHF Contributor Drexel28's Avatar
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    Quote Originally Posted by Krizalid View Post
    As for the second question, the integral obviously diverges for \alpha\le0, so the only interesting case to analyze convergence is \alpha>0. Here's a Hint: the integral converges for all \alpha>\frac12. Prove it.
    Doesn't then integral converges for all \alpha>1? \lim_{x\to\infty}\frac{\sin\left(\tfrac{1}{x^{\alp  ha}}\right)}{\tfrac{1}{x^{\alpha}}}=1. So that we can see that \int_1^{\infty}\sin\left(\tfrac{1}{x^{\alpha}}\rig  ht)dx shares convergence with \int_1^{\infty}\frac{dx}{x^{\alpha}}

    [Yes, it was a typo, it's fixed now. -K.]
    Last edited by Krizalid; January 2nd 2010 at 02:34 PM.
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