Why is the value of $\int_0^{2\pi}|2\cos(nx)+\sqrt{3}|\,dx$ independent of integer parameter $n$?

Question

I am not able to find an easy solution for the following formula

$$\int_0^{2\pi}|2\cos(nx)+\sqrt{3}|dx=4+\frac{4}{3}\pi\sqrt{3}.$$

Please help me prove it. Why it does not depend on the (positive) integer parameter $n$?

Answer 1

Outline: We address only the independence from the positive integer parameter $n$. Let $f(x)=|2\cos x+\sqrt{3}|$.

$1$) Show that for any $k$ between $0$ and $n-1$ $$\int_{2\pi k/n}^{2\pi(k+1)/n}f(nx)\,dx=\int_0^{2\pi/n} f(nx)\,dx.$$ This is done by a change of variable.

$2$) Show that $$\int_0^{2\pi/n} f(nx)\,dx=\frac{1}{n}\int_0^{2\pi} f(x)\,dx.$$ This is also done by a change of variable.

Remark: More informally, because of the periodicity of $f$, the integral breaks up naturally into a sum of $n$ equal parts. Each of the $n$ "areas" is $\frac{1}{n}$ of $\int_0^{2\pi}f(x)\,dx$, because we are scaling in the $x$-direction by the factor $\frac{1}{n}$.