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How do I solve this: $\int\limits_{0}^{1} \frac{\log(x^{2}-2\cdot x \cdot \cos{a}+1)}{x} \ dx$

Integration by parts is the only thing which I could think of, clearly that seems cumbersome. Substitution also doesn't work.

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    Yikes! How embarrassing. Thank you for the correction, Sasha.2011-08-15

3 Answers 3

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Assume $0. In this case $x^2-2 \, x \, \cos a+1 > 0$ for $0\le x \le 1$, so integral converges.

Denote $I(a)$ the value of the integral. Consider $\partial_a I(a)$. You can exchange the order of integration and differentiation, getting

$ \partial_a I(a) = \int_0^1 \mathrm{d} x \frac{2 \sin a}{x^2 + 1 - 2 \, x \, \cos a} = \pi - a $

The integral above is computed after a substitution $x = \cos a + \sin a \cdot \tan \frac{t}{2}$ and carefully considering integration region.

From here, knowing that for $I(\frac{\pi}{2}) = \frac{\pi^2}{24}$, we get

$I(a) = \frac{\pi^2}{24} + \int_{\frac{\pi}{2}}^a \mathrm{d} \alpha ( \pi - \alpha) = \pi a - \frac{a^2}{2} - \frac{\pi^2}{3}.$

The formula also extends to $a=0$ because the $I(0)$ converges.

The answer is extended to real $a$ by periodicity.

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    This is very nice and useful. +1. I have used this approach when I was teaching math at the polytechnic school. Too bad that in our first year course my teachers forgot to mention this. :)2011-08-15
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Rewriting the integral as $ \int\limits_{0}^{1} \frac{\log|x-e^{i a}|^2}{x} \ dx= \int\limits_{0}^{1} \frac{\log|1-xe^{-i a}|^2}{x} \ dx= $ $ \int\limits_{0}^{1} \frac{\log(1-xe^{i a })}{x} \ dx+ \int\limits_{0}^{1} \frac{\log(1-xe^{-i a })}{x} \ dx, $ expanding and integrating termwise leads to an answer in the form of series.

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    Also one can note that this series is the Fourier series of the answer.2011-08-15
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Please see $\textbf{Problem 4.30}$ in the book: ($\text{solutions are also given at the end}$)

  • $\text{The Math Problems Notebook,}$ by Valentin Boju and Louis Funar.