- #1
SithsNGiggles
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Homework Statement
I came across this integral recently while tutoring:
##\displaystyle \frac{1}{5} \int \frac{-x^3+2x^2-3x+4}{x^4-x^3+x^2-x+1}~dx##
Homework Equations
The Attempt at a Solution
I'm not sure how to approach this. At first I suspected partial fraction decomposition might be the way to go, but I don't know how to factor that denominator.
I've checked with WolframAlpha, and I'm getting a result that I haven't come across yet and don't quite understand: http://www.wolframalpha.com/input/?i=Integrate%5B%28-x%5E3%2B2x%5E2-3x%2B4%29%2F%28x%5E4-x%5E3%2Bx%5E2-x%2B1%29%2Cx%5D
##\displaystyle \int \cdots=\frac{1}{5}\sum_{\left\{\omega~:~\omega^4-\omega^3+\omega^2-\omega+1=0\right\}}
\frac{4\ln(x-\omega)-3\omega\ln(x-\omega)+2\omega^2\ln(x-\omega)-\omega^3\ln(x-\omega)}{4\omega^3-3\omega^2+2\omega-1}+C
##
How did WA get that answer? Is there some specific name for the method? (I've tried looking it up, but I can't quite put the question into words.)
From what I can tell, ##\omega## is an index indicating the roots of the polynomial in the denominator, or something like that.