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Math Help - integral of arctan(x^2) as a power series

  1. #1
    Junior Member pirateboy's Avatar
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    [Solved] integral of arctan(x^2) as a power series

    We're asked to show the improper integral
    \int \tan^{-1}(x^2)\,dx
    as a power series.

    I start with what I know. That is
    \displaystyle \tan^{-1}(x) = \sum_{n=0}^\infty (-1)^n \frac{x^{2n+1}}{2n+1} = x - \frac{1}{3}x^3 + \frac{1}{5}x^5 - \frac{1}{7}x^7 + \dots

    So...
    \displaystyle \tan^{-1}(x^2) = \sum_{n=0}^\infty (-1)^n  \frac{x^{4n+2}}{2n+1} = x^2 - \frac{1}{3}x^6 + \frac{1}{5}x^{10} -  \frac{1}{7}x^{14} + \dots

    I'm trying to see the pattern here, in order to put this in sigma notation, however I'm afraid that when I do I still won't grasp what's going on here with series.

    Knowing that, it's not hard to integrate the terms and see that

    \displaystyle \int\tan^{-1}(x)\,dx = C + \sum_{n=0}^\infty (-1)^n \frac{x^{2n+2}}{(2n+2)(2n+1)} = \frac{1}{2}x^2 - \frac{1}{14}x^4 + \frac{1}{30}x^6 - \frac{1}{56}x^8 + \dots

    But here's where I get stuck...
    \displaystyle \int \tan^{-1}(x^2)\,dx = C + \sum_{n=0}^\infty\text{???} = \frac{1}{3}x^3 - \frac{1}{21}x^7 + \frac{1}{55}x^{11} - \frac{1}{105}x^{15} + \dots

    Am I going about this the right way?

    Thanks in advance.
    Last edited by pirateboy; October 20th 2010 at 04:24 AM. Reason: term was wrong
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  2. #2
    MHF Contributor Also sprach Zarathustra's Avatar
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    \displaystyle \int \tan^{-1}(x^2)\,dx = C + \sum_{n=0}^\infty\text{???} = \frac{1}{3}x^3 - \frac{1}{21}x^7 + \frac{1}{55}x^{11} - \frac{1}{105}x^{15} + \dots
    I think that this is wrong.

    Integrate the formula of sum arctan(x^2) again...
    Last edited by Also sprach Zarathustra; October 20th 2010 at 04:04 AM.
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  3. #3
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    \displaystyle \tan^{-1}(x^2) = \sum_{n=0}^\infty (-1)^n \frac{x^{4n+\textbf{2}}}{2n+1}
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  4. #4
    Junior Member pirateboy's Avatar
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    oh, you're right. the first term is wrong in arctan(x^2). it should be x^2, not x. i'll fix it.
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  5. #5
    Junior Member pirateboy's Avatar
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    oh! that wasn't a copy and paste error. that was my problem. it was missing that 4n+2.

    thanks pandevil. i thinks i gots it nows.
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