MHB Log-sine and log-cosine integrals

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The discussion centers on proving specific integral identities involving logarithmic functions of sine and cosine over defined intervals. The integrals presented are: for \(0 \le \theta \le \pi\), \( \int_{0}^{\theta} \ln(\sin x) \ dx = - \theta \ln 2 - \frac{1}{2} \sum_{n=1}^{\infty} \frac{\sin (2n \theta)}{n^{2}}\), and for \(0 \le \theta \le \frac{\pi}{2}\), \( \int_{0}^{\theta} \ln(\cos x) \ dx = - \theta \ln 2 + \frac{1}{2} \sum_{n=1}^{\infty} (-1)^{n+1} \frac{\sin (2n \theta)}{n^{2}}\). Participants express interest in these results, with one suggesting a related problem involving logarithmic integrals with additional parameters. The conversation highlights the mathematical exploration of integrals and their properties. Overall, the thread emphasizes the significance of these logarithmic integral identities in mathematical analysis.
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For a few of you, this probably isn't very challenging. But I'm going to post it anyways since I find it interesting.
Show that for $0 \le \theta \le \pi$, $ \displaystyle \int_{0}^{\theta} \ln(\sin x) \ dx = - \theta \ln 2 - \frac{1}{2} \sum_{n=1}^{\infty} \frac{\sin (2n \theta)}{n^{2}}$.Also show that for $0 \le \theta \le \frac{\pi}{2}$, $ \displaystyle \int_{0}^{\theta} \ln(\cos x) \ dx = - \theta \ln 2 + \frac{1}{2} \sum_{n=1}^{\infty} (-1)^{n+1} \frac{\sin (2n \theta)}{n^{2}}$.
 
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I quite agree, RV... Very interesting! (Heidy)I'll not answer that one for reasons we both understand, but if it's not entirely impertinent of me - which it might well be :o - I'd like to propose the following one for you... Something to get you teeth into.For $$a,\, b,\, c > 0 \in \mathbb{R}$$, $$m\in \mathbb{Z}^+\ge 1$$, and $$0 < \theta \le \pi/2$$, find the additional conditions on the parameters as well as the closed form solution for:$$\int_0^{\theta}\log^m(a+b\cos x +c\sin x)\,dx$$
 
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