1no
what u have found is
I(0),what i am asking is for any general n
\int \frac{dx}{2-\cos x}=\int \frac{dx}{1+2\sin^2 x/2}=\int \frac{dx}{\cos^2 x/2 +3\sin^2x/2}=2\int \frac{d(\tan \frac{x}{2})}{1+(\sqrt{3}\tan(\frac{x}{2})^2)}\\=\frac{2}{\sqrt{3}}\tan^{-1}(\sqrt{3}tan(\frac{x}{2}))+C
without using hi fi concepts we get I(0)=π/√3