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Math Help - rational integral

  1. #1
    Junior Member
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    rational integral

    Hi, can somebody give any hint how to get this integral \int\frac{x^3}{x^4+x^3+x^2+x+1}dx ?
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  2. #2
    MHF Contributor chisigma's Avatar
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    Is...

    1 + x + x^{2} + x^{3} + x^{4} = \prod_{k=1}^{4} (x-x_{k}) , x_{k} = e^{\frac{2 \pi k i}{5}} (1)

    ... so that...

    \frac{x^{3}}{1 + x + x^{2} + x^{3} + x^{4}}= \sum_{k=1}^{4} \frac{r_{k}}{(x-x_{k})} (2)

    ... where...

    r_{k} = \lim_{x \rightarrow x_{k}} \frac {x^{3}\cdot (x-x_{k})}{1 + x + x^{2} + x^{3} + x^{4}} = \frac{x_{k}^{3}}{1 + 2 x_{k} + 3 x_{k}^{2} + 4 x_{k}^{3}} (3)

    ... so that is...

    \int \frac{x^{3}}{1 + x + x^{2} + x^{3} + x^{4}} dx = \sum_{k=1}^{4} r_{k} \cdot \ln (x-x_{k}) + c (4)

    Kind regards

    \chi \sigma
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