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Hi when studying a physical problem i have to evaluate these three integrals:

$$\int_{0}^{\pi}\ln(\cos(x)+1)\,dx$$

$$\int_{0}^{\pi}\ln(\cos(x)+1)\cos(nx)\,dx$$

$$\int_{0}^{\pi}\ln(\cos(x)+1)\sin(nx)\,dx$$

Where $n$ is a positive integer.

I found the first one: $-\pi \ln(2)$, but I'm unable to find the other two, any ideas?

egreg
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Jonat
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  • Can it be related to the Fourier series? –  Dec 06 '16 at 12:51
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    Hint: Use the identity $\cos(x) + 1 = 2\cos^2(x/2)$ and the Fourier representation of $\log(\cos(x))$: http://math.stackexchange.com/questions/292468/fourier-series-of-log-sine-and-log-cos – Dominik Dec 06 '16 at 12:55

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$$\begin{eqnarray*}\int_{0}^{\pi}\log(1+\cos\theta)\cos(n\theta)\,d\theta &=& \frac{1}{n}\int_{0}^{\pi}\frac{\sin(n\theta)\sin(\theta)}{1+\cos\theta}\,d\theta\\(\theta\mapsto 2\varphi)\quad&=&\frac{1}{n}\int_{0}^{\pi/2}\frac{2\sin(2n\varphi)\sin(\varphi)}{\cos(\varphi)}\,d\varphi\\(\text{induction})\quad&=&\frac{(-1)^{n+1}\pi}{n}\end{eqnarray*}$$

$$\begin{eqnarray*}\int_{0}^{\pi}\log(1+\cos\theta)\sin(n\theta)\,d\theta &=& \log(2)\int_{0}^{\pi}\sin(n\theta)\,d\theta+4\int_{0}^{\pi/2}\log(\cos\varphi)\sin(2n\varphi)\,d\varphi\\&=&\frac{1-(-1)^n}{n}\log 2+4(-1)^{n+1}\int_{0}^{\pi/2}\frac{(1-\cos(2n\varphi))\cos\varphi}{\sin\varphi}\,d\varphi\end{eqnarray*}$$ and by induction: $$\int_{0}^{\pi/2}\frac{\sin^2(n\varphi)\cos\varphi}{\sin\varphi}\,d\varphi = \underbrace{\frac{1}{2}+\frac{1}{2}+\frac{1}{6}+\frac{1}{6}+\frac{1}{10}+\frac{1}{10}+\frac{1}{14}+\ldots}_{n \text{ terms}}$$

Jack D'Aurizio
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