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I've been struggling with this integral for a few days.

$$\int_{0}^{\pi /2}\!\arccos\left(\frac{\cos(x)}{1+2\cos(x)}\right)dx$$

Any help would be appreciated. Cheers.

Kenta S
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Lmnop
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1 Answers1

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This is the so-called Coxeter integral. It is related to Ahmed integrals. You can refer to this and this for calculating it. The result is $$ \int_{0}^{\frac{\pi}{2}}\arccos \left (\frac{ \cos(x)}{1+2 \cos(x)}\right )dx=\frac{5\pi^2}{24} $$

Also you can find more on this page.

FFjet
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    I saw solution of the integral. The first step is to simplify the integrand and then maybe integration by parts — but Feynman algorithm shall definitely be used in order to compute integral. I found the document wherein I discerned solution but can't attach so you would see it. Integral is rather simple but from the point of technical view :) – sergei ivanov Dec 20 '20 at 15:28
  • Search in Google: Two very challenging calculus problems. You will, should you have not yet, understand about what I am speaking of. – sergei ivanov Dec 20 '20 at 15:30
  • Could you please put a link here? I don't understand what you mean. – FFjet Dec 20 '20 at 15:38
  • https://www.math.ucla.edu/~josephbreen/Some_Very_Challenging_Calculus_Problems-2.pdf – Lmnop Dec 20 '20 at 16:21
  • @Lmnop OK, but the series was solved by Mathematica in $0.0003$ sec, while the integral... no way! – Raffaele Dec 20 '20 at 17:13