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Math Help - Integral as a power series

  1. #1
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    Integral as a power series

    Hey everyone,

    I do not know how to express this integral as a power series:

    \int{\frac{x-arctan(x)}{x^3}


    I cannot figure out how to get this into the right form. I am a bit confused. Any help is appreciated.
    Thanks
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  2. #2
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    Quote Originally Posted by evant8950 View Post
    Hey everyone,

    I do not know how to express this integral as a power series:

    \int{\frac{x-arctan(x)}{x^3}


    I cannot figure out how to get this into the right form. I am a bit confused. Any help is appreciated.
    Thanks
    \displaystyle \arctan{x} = x - \frac{x^3}{3} + \frac{x^5}{5} - \frac{x^7}{7} + ...

    \displaystyle x - \arctan{x} = \frac{x^3}{3} - \frac{x^5}{5} + \frac{x^7}{7} - \frac{x^9}{9} +  ...<br />

    \displaystyle \frac{x - \arctan{x}}{x^3} = \frac{1}{3} - \frac{x^2}{5} + \frac{x^4}{7} - \frac{x^6}{9} +  ...<br />

    integrate the last series.
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  3. #3
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    Quote Originally Posted by skeeter View Post
    \displaystyle \arctan{x} = x - \frac{x^3}{3} + \frac{x^5}{5} - \frac{x^7}{7} + ...

    \displaystyle x - \arctan{x} = \frac{x^3}{3} - \frac{x^5}{5} + \frac{x^7}{7} - \frac{x^9}{9} +  ...<br />

    \displaystyle \frac{x - \arctan{x}}{x^3} = \frac{1}{3} - \frac{x^2}{5} + \frac{x^4}{7} - \frac{x^6}{9} +  ...<br />

    integrate the last series.

    I understand that the \int{\frac {1}{1+x^2} is equal to arctanx. Then I converted that to a power series and integrated \sum_{n=0}^\infty\((-1)^n\frac {x^{2n +1}}{2n+1}}+C

    Where do I go from here?
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  4. #4
    MHF Contributor Also sprach Zarathustra's Avatar
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    Read again the post above...

    1. Replace the arctan(x) function in your integral with power series of arctan(x)

    2. Understand what skeeter wrote.

    3. Integrate term-by-term (Why you allowed to do this?)
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  5. #5
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    I understand #2 on what skeeter wrote. I understand on #3 when the x^3 is multiplied back. What happened to the x in #2? I see the negative got distributed, but why is x not being added to each term?

    Thanks
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  6. #6
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    Quote Originally Posted by evant8950 View Post
    I understand #2 on what skeeter wrote. I understand on #3 when the x^3 is multiplied back. What happened to the x in #2? I see the negative got distributed, but why is x not being added to each term?

    Thanks
    why would x be added to each term? ... just subtract the series for arctan(x) from x.

    \displaystyle x - \arctan{x} = x - \left(x - \frac{x^3}{3} + \frac{x^5}{5} - \frac{x^7}{7} + ... \right) = x - x + \frac{x^3}{3} - \frac{x^5}{5} + \frac{x^7}{7} - \frac{x^9}{9} +  ... \, = \frac{x^3}{3} - \frac{x^5}{5} + \frac{x^7}{7} - \frac{x^9}{9} +  ...
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  7. #7
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    Thanks! I got it now.
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