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I have seen integrals of the form $$\int \frac{1}{ax+b}dx$$ and $$\int \frac{1}{ax^{2}+bx+c} dx$$ But I cannot see how to integrate reciprocals of higher degree - does there exist a general solution to the integrals of reciprocals of cubics, quartics, and higher?

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Reciprocals of higher degree can have their denominators factored into linear and/or irreducible quadratic terms, and from there, our result can be obtained through partial fraction decomposition.

For more details, see Arturo's excellent answer to this question.

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    $\begingroup$ Note that just because we know a partial-fraction decomposition exists doesn't mean that we can explicitly express the coefficients of that decomposition (any better than 'the $n$th root of the polynomial $p(x)$', at least); this is a consequence of Galois' famous result that polynomials of degree higher than 4 may have solutions not expressible through radicals. So we can say what form the integral will take, but not necessarily explicitly express the result 'elementarily'. $\endgroup$ – Steven Stadnicki Sep 18 '13 at 21:37
  • $\begingroup$ That's a good point, Steven. $\endgroup$ – Cameron Buie Sep 18 '13 at 23:05
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Assume the coefficients of the polynomial $f(x)$ are real: if they're not, the question will be answered quite differently.

Every root of the polyomial must then either be real or part of a pair of complex conjugates $a\pm bi$, where $a$ and $b$ are real.

Then $$ f(x) = c(x-\bullet)(x-\bullet)(x-\bullet)\cdots(x-\bullet) $$ where $c$ is the leading coefficient and each "$\bullet$" is one of the roots. If you get a real root, you've got a first-degree factor $(x-\bullet)$. If you get a pair of conjugates, then you have something like $$ (x - (3+5i)) (x - (3-5i)). $$ When you multiply this out, the imaginary parts cancel: $$ (x - (3+5i)) (x - (3-5i)) = x^2 - 3x - 5ix -3x + 5ix + 9 + 15i - 15i + 25 $$ $$ = 3x^2 - 6x + 34. $$ There you have a quadratic factor.

So you just get first-and second-degree factors.

(Finding just what those factors are, in the case of, e.g. a 15th-degree polynomial, can be quite a lot of work.)

How do we know that $f(x)$ factors as $$ c(x-\bullet)(x-\bullet)(x-\bullet)\cdots(x-\bullet)\ ? $$ That goes back to Carl Gauss in the year 1799. It is sometimes called the fundamental theorem of algebra, a name that some people object to on the grounds that it's a misnomer.

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  • $\begingroup$ how can we for example factor x^4+1 into 1st and 2nd degree since it's factors are $((-1)^{(1/4)}-x) ((-1)^{(3/4)}-x) (x+(-1)^{(1/4)}) (x+(-1)^{(3/4)})$ $\endgroup$ – kiyarash Oct 8 '15 at 21:28
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    $\begingroup$ since when we multiply conjugates $((-1)^{(1/4)}-x)$ and $ (x+(-1)^{(1/4)})$ we get $-x^2+i$ $\endgroup$ – kiyarash Oct 8 '15 at 21:32

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