Answer
$$\int_{0}^{2 \pi} \sqrt{\frac{1-\cos x}{2}} dx =4$$
Work Step by Step
Given $$\int_{0}^{2 \pi} \sqrt{\frac{1-\cos x}{2}} dx $$
So, we have
\begin{aligned}
I&= \int_{0}^{2 \pi} \sqrt{\frac{1-\cos x}{2}} \ dx\\
&\text{since} \ \ \cos x= 1-2\sin^2\frac{ x}{2} \Rightarrow \sin ^{2} \frac{x}{2}=\frac{1-\cos x}{2},\text{ we get}\\
I&=\int_{0}^{2 \pi}\left|\sin \frac{x}{2}\right| dx\\
&=2\int_{0}^{2 \pi} \frac{1}{2}\sin \frac{x}{2} \ d x\\
&=\left[-2 \cos \frac{x}{2}\right]_{0}^{2 \pi}\\
&=\left[-2 \cos\pi\right]- \left[-2 \cos0\right]\\
&=2+2\\
&=4\\
\end{aligned}
