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Math Help - Integral that I just can't figure out !

  1. #1
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    Question Integral that I just can't figure out !

    Hi all, for some reason I just keep going round in circles with this indefinite integral;

    Int: (2x^3 + x) * arctan[x] dx

    Sorry for lack of formatting. This is driving me mad, I know it should be jumping out at me what to do.

    Thanks in advance for your help !!!

    Stu
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  2. #2
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    Re: Integral that I just can't figure out !

    \displaystyle \begin{align*} \int{\left( 2x^3 + x \right) \arctan{(x)}\,dx} &= \left( \frac{x^4}{2} + \frac{x^2}{2} \right) \arctan{(x)} - \int{ \left( \frac{x^4}{2} + \frac{x^2}{2} \right)\left( \frac{1}{1 + x^2} \right) dx } \\ &= \frac{x^2}{2}\left( x^2 + 1 \right) \arctan{(x)} - \frac{1}{2} \int{ \frac{x^2 \left( x^2 + 1 \right)}{1 + x^2}\, dx } \\ &= \frac{x^2}{2} \left( x^2 + 1 \right) \arctan{(x)} - \frac{1}{2} \int{ x^2\,dx } \\ &= \frac{x^2}{2} \left( x^2 + 1 \right) \arctan{(x)} - \frac{x^3}{6} + C \end{align*}
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    Re: Integral that I just can't figure out !

    Thankyou
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