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Thread: Help needed in solving Integration and Convergence Test

  1. #1
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    Help needed in solving Integration and Convergence Test

    integrate: $\displaystyle [sqrt(X^2 + 6)] / X$

    Determine the convergence :
    $\displaystyle [(-1)N!]/N^N$

    Help me pls... thank u...
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  2. #2
    Senior Member Peritus's Avatar
    Joined
    Nov 2007
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    $\displaystyle
    \begin{gathered}
    u = \sqrt {x^2 + 6} \hfill \\
    du = \frac{x}
    {{\sqrt {x^2 + 6} }}dx \hfill \\
    \end{gathered} $

    $\displaystyle
    \int {\frac{{u^2 }}
    {{u^2 - 6}}du} = \int {1 + \frac{6}
    {{\left( {u + 6} \right)\left( {u - 6} \right)}}du = u + \frac{1}
    {2}\int {\frac{1}
    {{u - 6}} - \frac{1}
    {{u + 6}}} } du
    $

    $\displaystyle
    = u + \frac{1}
    {2}\left[ {\ln \left| {u - 6} \right| - \ln \left| {u + 6} \right|} \right] = \sqrt {x^2 + 6} + \ln \sqrt {\frac{{\sqrt {x^2 + 6} - 6}}
    {{\sqrt {x^2 + 6} + 6}}}

    $

    to determine the convergence use can use the ratio test
    Last edited by Peritus; Oct 26th 2008 at 09:47 AM.
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  3. #3
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    Thank you very much...
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