# How do I proceed Calculating. $\int_0^{\frac{\pi}{2}} \arccos ( \frac{\cos x}{1+2 \cos x}) dx$ [closed]

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.....

• what have u tried? Any own thoughts... – tired Dec 4 '15 at 15:04
• Looks like $\frac{5\pi^2}{24}$... – Vladimir Reshetnikov Dec 4 '15 at 20:41
• It also appears that $\displaystyle\int_0^{\pi/3} \arccos \left( \frac{\cos x}{1+2 \cos x}\right) dx=\frac{2\pi^2}{15},\,\int_0^{\pi/5} \arccos \left( \frac{\cos x}{1+2 \cos x}\right) dx=\frac{71\pi^2}{900},\,\int_0^{3\pi/5} \arccos \left( \frac{\cos x}{1+2 \cos x}\right) dx=\frac{241\pi^2}{900}.$ – Vladimir Reshetnikov Dec 4 '15 at 21:15
• I can't even think how to proceed – Aditya Dec 5 '15 at 2:36
• @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. – David H Dec 6 '15 at 7:20