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I am trying to compute the following integral: $$\int_0^{2\pi} \frac{y}{y^n-1} dy$$

I've tried to decompose $y^n-1$ into $(y-1)(y-e^{i\theta})(y-e^{i2\theta})...(y-e^{i(n-1)\theta})$ but I don't know what to do with this factorization. I've read some others similar questions with the answers but I don't know if the same methods apply.

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    You are integrating on an interval and I am not sure what the roots of unity have do with anything. Also, the integral doesn't exist.2011-06-30
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    That is weird. I was originally trying to integrate $\int_{S^1} dz/(z^n-a)$ and that's how I got this integral (I didn't include the factor $a^{2/(n-1)}$ that I got from the substitution I did for simplicity). $S^1$ is the unit circle.2011-06-30
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    OK. Assume $|a|\not=1$. Treat the cases $|a|>1$ and $|a|<1$ separately. One case is easy. Try the other case by thinking Cauchy's Integral formula, small circles around the poles and a suitable contour that incorporates these small circles and the unit circle.2011-06-30
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    Thank you! I'll come back if I still can't make substantial progress.2011-06-30
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    The integral in your question isn't the integral you want. You want to replace $y$ with $z = e^{iy}$ in the integrand or something similar.2011-06-30
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    The integral in the body of your question has a pole at $1$, and your path of integration goes through $1$, so it is undefined. However, as Qiaochu says, it sounds like this isn't the integral you wanted.2011-07-01
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    OK, I must have done a mistake during the substitution.2011-07-01

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