# Integration Problem: $\int \frac{x^2+1}{x^5-1}dx$

$$\int \frac{x^2+1}{x^5-1}dx$$

I am unable to integrate it, nothing works. Yes, I can use partial fraction but who remembers factorization of $x^5-1$, I need a better way of doing this.

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You don't have to "remember" the factorization of $x^5-1$. Notice that $x=1$ is a solution and use long division to see that $x^5-1=(x-1)(x^4+x^3+x^2+x+1)$. – chris Apr 14 '12 at 6:55
@chris: Long division? Geometric sum formula bro. – anon Apr 14 '12 at 6:56
@anon: That works too. It's useful to know that, in general, if you can spot a solution then long division is a way to get the factorization of the polynomial though. – chris Apr 14 '12 at 6:57
If this is homework problem, you have a cruel instructor. (Unless there is a smart way of doing this that I haven't figured out yet) – Shitikanth Apr 14 '12 at 7:13
Very cruel indeed; this results in an unholy mass of arctangents and logarithms... – J. M. Apr 14 '12 at 7:47

Take the denominator, find the 5 complex roots,one of them is 1.Descompose the denominator as: $$(x^5 - 1) = (x-1)\cdot(x-a)(x-a^*)\cdot(x-b)(x-b^*)$$ Remember that $(x-a)(x-a^*)$ is a polynomial of second degree with real coefficients. Then apply descomposition in simple (partial) fractions.

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+1: In case the OP or somebody else has problems finding the roots, they are the fifth primitive roots of unity: $a=e^{2\pi i/5}$, $b=e^{4\pi i/5}$. Here $$(x-a)(x-a^*)=x^2-(a+a^*)x+aa^*=x^2-(2\cos\frac{2\pi}5)x+1.$$ Remember that $$2\cos\frac{2\pi}5=\frac{\sqrt{5}-1}2.$$ The coefficients become a bit cumbersome, but this allows the use of the common techniques of finding the indefinite integral of a rational function. – Jyrki Lahtonen Apr 14 '12 at 7:53

There's always this solution that can be used in a neighborhood of $0$ with a convergence radius of $1$ : $$\int \frac{x^2 + 1}{x^5-1} \, dx = \int -(x^2+1) \left( \sum_{i=0}^{\infty} x^{5i} \right) \, dx = \sum_{i=0}^{\infty} \frac{x^{5i+1}}{5i+1} + \sum_{i=0}^{\infty} \frac{x^{5i+3}}{5i+3}$$ but unless that's what you were expecting, I don't thing it's worth very much. Note that the expansion alpha.Debi was suggesting will probably involve at some point the integration of terms of the form $1/(x-a)$, which will most probably bring up logarithms, which are not well-behaving (in the sense that they are multivalued functions over $\mathbb C$) for integration with respect to a path (even though it does work, I'm just mentioning "there's a point to notice there"). Perhaps the solution in terms of logarithms obtained in this fashion is equivalent to this one in a neighborhood of $0$.

Hope that helps,

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But in the descomposition I suggested , we forget the complex numbers after multiplying (x-a)(x-conjugate a),then we continue only in the way of integral of simple fractions all with real coefficients. – alpha.Debi Apr 14 '12 at 7:51

I'd like to point out that you can factorize $x^5 -1$ quite easily by observing that you are actually finding the fifth roots of 1: $$x =\sqrt[5]{1} = e^{2\pi i k/5}$$ So that as @alpha.Debi pointed out, factorizing the denominator may be long, but is quit straight forward.

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For these cyclotomic denominators, one can get an answer pretty fast by factoring into roots of unity. Let $\omega_k=e^{i\theta_k}=\cos\theta_k+i\sin\theta_k$, where $\theta_k=2\pi k/5$. Then $$\frac{x^2+1}{x^5-1}=\frac{x^2+1}{\prod_{k=0}^4(x-\omega_k)}=\sum_{k=0}^4\frac{A_k}{x-\omega_k}$$ Obeserving that $\omega_k^{-k}=\omega_k^{5-k}$ and applying L'Hopital's rule a couple of times, \begin{align}\lim_{x\rightarrow\omega_k}\frac{(x^2+1)(x-\omega_k)}{x^5-1} & =\lim_{x\rightarrow\omega_k}\frac{(x^2+1)(1)}{5x^4}=\frac15\left(\omega_k^{-2}+\omega_k\right) \\ & =\lim_{x\rightarrow\omega_k}\sum_{j=0}^4\frac{A_j\left(x-\omega_k\right)}{x-\omega_j}=\sum_{j=0}^4A_j\delta_{kj}=A_k\end{align} Then \begin{align}\int\frac{x^2+1}{x^5-1}dx & =\sum_{k=0}^4A_k\int\frac{dx}{x-\omega_k}=\sum_{k=0}^4\frac15\left(\omega_k^{-2}+\omega_k\right)\ln\left(x-\omega_k\right)+C_1 \\ & =\sum_{k=0}^4\left(\cos2\theta_k+\cos\theta_k-i\sin2\theta_k+i\sin\theta_k\right)\ln\left(x-\cos\theta_k-i\sin\theta_k\right)+C_1 \\ & =\sum_{k=0}^4\left(\cos2\theta_k+\cos\theta_k-i\sin2\theta_k+i\sin\theta_k\right)\left\{\frac12\ln\left(x^2-2x\cos\theta_k+1\right)+i\,\text{atan2}\left(-\sin\theta_k,x-\cos\theta_k\right)\right\}+C_1 \\ & =\sum_{k=0}^4\left(\cos2\theta_k+\cos\theta_k-i\sin2\theta_k+i\sin\theta_k\right)\left\{\frac12\ln\left(x^2-2x\cos\theta_k+1\right)+i\,\text{atan2}\left(x-\cos\theta_k,\sin\theta_k\right)\right\}+C \\ & = \sum_{k=0}^4\left\{\frac12\left(\cos2\theta_k+\cos\theta_k\right)\ln\left(x^2-2x\cos\theta_k+1\right)+\left(\sin2\theta_k-\sin\theta_k\right)\tan^{-1}\left(\frac{x-\cos\theta_k}{\sin\theta_k}\right)\right\}+C\end{align} because the imaginary parts cancel out above. The $k=0$ term is just $\frac25\ln|x-1|$ and the $k=4$ term is a copy of the $k=1$ term, just as the $k=3$ term is a copy of the $k=2$ term. So substituting in the values of the trig functions above and simplifying, we get \begin{align}\int\frac{x^2+1}{x^5-1}dx=\frac25\ln|x-1| & -\frac1{10}\ln\left(x^2+\frac{1-\sqrt5}2x+1\right) \\ & +\frac{\sqrt{10-2\sqrt5}\left(1-\sqrt5\right)}{20}\tan^{-1}\left(\frac{4x-\sqrt5+1}{\sqrt{10+2\sqrt5}}\right) \\& -\frac1{10}\ln\left(x^2+\frac{1+\sqrt5}2x+1\right) \\ & -\frac{\sqrt{10+2\sqrt5}\left(1+\sqrt5\right)}{20}\tan^{-1}\left(\frac{4x+\sqrt5+1}{\sqrt{10-2\sqrt5}}\right)+C\end{align}

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