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?
Integrating Reciprocals of Polynomials
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0possible duplicate o$f$: http://math.stackexchan$g$e.com/questions/20963/integration-by-partial-fractions-how-and-why-does-it-work – 2012-06-06
2 Answers
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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0That's a good point, Steven. – 2013-09-18
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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1since when we multiply conjugates $((-1)^{(1/4)}-x)$ and $ (x+(-1)^{(1/4)})$ we get $-x^2+i$ – 2015-10-08