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How do I proceed to integrate $$ \int_0^{\frac{\pi}{2}} \arccos \left( \frac{\cos x}{1+2 \cos x}\right) dx $$ The question is complete..looking forward for any help.....

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  • $\begingroup$ what have u tried? Any own thoughts... $\endgroup$ – tired Dec 4 '15 at 15:04
  • $\begingroup$ Looks like $\frac{5\pi^2}{24}$... $\endgroup$ – Vladimir Reshetnikov Dec 4 '15 at 20:41
  • $\begingroup$ It also appears that $\displaystyle\int_0^{\pi/3} \arccos \left( \frac{\cos x}{1+2 \cos x}\right) dx=\frac{2\pi^2}{15},\,\int_0^{\pi/5} \arccos \left( \frac{\cos x}{1+2 \cos x}\right) dx=\frac{71\pi^2}{900},\,\int_0^{3\pi/5} \arccos \left( \frac{\cos x}{1+2 \cos x}\right) dx=\frac{241\pi^2}{900}.$ $\endgroup$ – Vladimir Reshetnikov Dec 4 '15 at 21:15
  • $\begingroup$ I can't even think how to proceed $\endgroup$ – Aditya Dec 5 '15 at 2:36
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    $\begingroup$ @ADITYA This looks like one of Coxeter's integrals. These can be tricky to solve from scratch, but fortunately there is a lot of information about them on the Internet. $\endgroup$ – David H Dec 6 '15 at 7:20