1. **Problem statement:** Evaluate the integral $$I = \int \sqrt{1 + \cos x} \, dx$$. 2. **Formula and trigonometric identity:** Use the half-angle identity for cosine: $$1 + \cos x = 2 \cos^2 \left(\frac{x}{2}\right)$$. 3. **Rewrite the integral:** Substitute the identity into the integral: $$I = \int \sqrt{2 \cos^2 \left(\frac{x}{2}\right)} \, dx = \int \sqrt{2} \left|\cos \left(\frac{x}{2}\right)\right| \, dx$$. Assuming $$\cos \left(\frac{x}{2}\right) \geq 0$$ in the interval considered, we simplify to: $$I = \sqrt{2} \int \cos \left(\frac{x}{2}\right) \, dx$$. 4. **Integration:** Use the substitution formula: $$\int \cos(ax) \, dx = \frac{\sin(ax)}{a} + C$$. Here, $$a = \frac{1}{2}$$, so: $$I = \sqrt{2} \cdot \frac{\sin \left(\frac{x}{2}\right)}{\frac{1}{2}} + C = 2 \sqrt{2} \sin \left(\frac{x}{2}\right) + C$$. 5. **Final answer:** $$\boxed{I = 2 \sqrt{2} \sin \left(\frac{x}{2}\right) + C}$$ This completes the evaluation of the integral using the half-angle formula and basic integration rules.