The given integral is:

$I = \int_0^{2\pi} \sqrt{\frac{1 + \cos x}{2}} \, dx$

To solve this, let's proceed step-by-step.

Step 1: Simplify the Expression

We recognize that $\frac{1 + \cos x}{2}$ is the half-angle identity for $\cos^2\left(\frac{x}{2}\right)$, which means:

$\frac{1 + \cos x}{2} = \cos^2\left(\frac{x}{2}\right)$

Thus, the integral becomes:

$I = \int_0^{2\pi} \sqrt{\cos^2\left(\frac{x}{2}\right)} \, dx$

Step 2: Simplify the Square Root

Since $\sqrt{\cos^2\left(\frac{x}{2}\right)} = |\cos\left(\frac{x}{2}\right)|$, the integral becomes:

$I = \int_0^{2\pi} |\cos\left(\frac{x}{2}\right)| \, dx$

Step 3: Break the Integral at Critical Points

We know that $\cos\left(\frac{x}{2}\right)$ changes sign at $x = \pi$. Therefore, we split the integral into two parts:

$I = \int_0^\pi \cos\left(\frac{x}{2}\right) \, dx + \int_\pi^{2\pi} -\cos\left(\frac{x}{2}\right) \, dx$

Step 4: Solve Each Integral

Let's solve each part separately.

For the first part:

$\int_0^\pi \cos\left(\frac{x}{2}\right) \, dx = 2 \sin\left(\frac{x}{2}\right) \Big|_0^\pi = 2$

For the second part:

$\int_\pi^{2\pi} -\cos\left(\frac{x}{2}\right) \, dx = -2 \sin\left(\frac{x}{2}\right) \Big|_\pi^{2\pi} = 2$

Step 5: Add the Results

Adding both integrals:

$I = 2 + 2 = 4$

Thus, the value of the integral is:

$I = 4$

The correct answer is C. 4.

Do you need any additional details or have any questions?

Here are 5 related questions:

1. What is the geometric interpretation of $\cos x$ in trigonometry?
2. How do you apply trigonometric identities in integration?
3. What is the role of absolute value in this integral and why is it necessary?
4. Can this integral be generalized to other trigonometric functions?
5. How does the integral change if the limits of integration are altered?

Tip: Always check if trigonometric identities can simplify an integral before solving it, as it often leads to easier computations.