Find the integral
$$\int_0^{\frac{\pi}{2}}\arccos\left(\frac{\cos(x)}{1+2\cos(x)}\right)~\mathrm{d}x.$$
Been struggling on this one for ages.
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You should post some of that struggle so we can help you in more detail. – eyeballfrog Dec 15 '22 at 16:38
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Well, I tried making a substitution for the part in the brackets, but the result was something long and complicated – rmak Dec 15 '22 at 16:46
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1https://math.stackexchange.com/questions/4597839/ would be a similar question. It is interesting to investigate the differences between these questions. – Kazuki OKAMURA Dec 15 '22 at 16:46
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2This one's famous. – J.G. Dec 15 '22 at 17:02
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You may try $y=\cos^{-1}\left(\frac{\cos(x)}{1-2\cos(x)}\right)$ to get rid of the inverse cosine – Тyma Gaidash Dec 15 '22 at 17:02
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Numeric evaluation gives $5\pi^2/24$. Possible zeta function involvement. – eyeballfrog Dec 15 '22 at 17:03
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3Further to my previous comment, here is a concise summary of the "monster" calculation. – J.G. Dec 15 '22 at 17:07