The supremum of a sequence of definite integrals I am interested to find the supremum of the following sequence of definite integrals:
$$
I_n=\int_0^\pi\sqrt{4\cos^2((2n+1)x)+4\cos((2n+1)x)\cos(x)+1}\,\textrm{d}x,\ n\ge 1.
$$
One of my ideas was to use the inequality $\cos^2(x)\le 1$ in order to obtain the following majorization:
$$
I_n\ge\int_0^\pi|2\cos((2n+1)x)\cos(x)+1|\,\textrm{d}x.
$$
Unfortunately, this majorization is not as strong as necessary.

I managed to show that $I_n$ indeed converges.
\begin{align}
J_n &= \int_{0}^{\pi}\sqrt{4\cos^2 nx+4\cos nx\cos x+1}\,\textrm{d}x \\
&=\frac{1}{n}\sum_{k=0}^{n-1}\int_{k\pi}^{(k+1)\pi}\sqrt{4\cos^2 x+4\cos x\cos\frac{x}{n}+1}\,\textrm{d}x \\
&=\frac{1}{n}\sum_{k=0}^{n-1}\int_{k\pi}^{(k+1)\pi}\sqrt{4\cos^2 x+4\cos x\cos\frac{k\pi}{n}+1}\,\textrm{d}x+O\left(\frac{1}{n}\right) \\
&=\int_{0}^{\pi}\frac{1}{n}\sum_{k=0}^{n-1}\sqrt{4\cos^2 x+4\cos x\cos\frac{k\pi}{n}+1}\,\textrm{d}x+O\left(\frac{1}{n}\right) \\
&=\int_{0}^{\pi}\int_0^1\sqrt{4\cos^2 x+4\cos x\cos\pi y+1}\,\textrm{d}y\,\textrm{d}x+O\left(\frac{1}{n}\right)
\end{align}
So as $n\rightarrow \infty$, $$I_n\rightarrow \frac{1}{\pi}\int_{0}^{\pi}\int_0^\pi\sqrt{4\cos^2 x+4\cos x\cos y+1}\,\textrm{d}y\,\textrm{d}x=4.946...$$
