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Math Help - Solve Integral Using Series

  1. #1
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    Solve Integral Using Series

    The problem is the Integrand of e^(4t)/(e^(2t)+3*e^(t)+2)

    Any ideas on how to begin would be greatly appreciated.
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  2. #2
    MHF Contributor chisigma's Avatar
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    Setting e^{t}=x the function you have to integrate becomes...

     \frac {e^{4t}}{e^{2t}+3\cdot e^{t} + 2} = \frac{x^{4}}{x^{2} + 3\cdot x + 2} = \frac{x^{4}}{(1+x)\cdot (2+x)} (1)

    Now is...

    \frac{1}{(1+x)\cdot (2+x)}= \frac{1}{1+x} -\frac{1}{2}\cdot \frac{1}{1+\frac{x}{2}} (2)

    ... and the two fractional terms in (2) can be written as series in the following manner...

    \frac{1}{1+x} = 1 - x + x^{2} - x^{3} + \dots

    \frac{1}{1+\frac{x}{2}} = 1 - \frac {x}{2} + (\frac{x}{2})^{2} - (\frac{x}{2})^{3} + \dots (3)

    Now you insert (3) in (2), then multiply by x^{4} and obtain (1) , then set  x= e^{t} in (1) and [finally!...] at this point you can integrate 'term by term'... a little tedious job ...

    Kind regards

    \chi \sigma

    A little p.s. : the series expansions (3) are possible only for |x|<1, so that the procedure is correct only for |e^{t}|<1, i.e. only for t<0...
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