1. **State the problem:** We need to evaluate the integral $$\int \sqrt{1+\cos x} \, dx$$. 2. **Use a trigonometric identity:** Recall that $$1+\cos x = 2\cos^2\left(\frac{x}{2}\right)$$. 3. **Rewrite the integral:** Substitute the identity into the integral: $$\int \sqrt{1+\cos x} \, dx = \int \sqrt{2\cos^2\left(\frac{x}{2}\right)} \, dx = \int \sqrt{2} \left|\cos\left(\frac{x}{2}\right)\right| \, dx$$. 4. **Simplify the integral:** Assuming the domain where $$\cos\left(\frac{x}{2}\right) \geq 0$$, we can drop the absolute value: $$= \sqrt{2} \int \cos\left(\frac{x}{2}\right) \, dx$$. 5. **Integrate:** Use substitution: Let $$u = \frac{x}{2}$$, so $$du = \frac{1}{2} dx$$ or $$dx = 2 du$$. Then, $$\sqrt{2} \int \cos(u) \cdot 2 \, du = 2\sqrt{2} \int \cos u \, du = 2\sqrt{2} \sin u + C$$. 6. **Back-substitute:** Replace $$u$$ with $$\frac{x}{2}$$: $$2\sqrt{2} \sin\left(\frac{x}{2}\right) + C$$. **Final answer:** $$\int \sqrt{1+\cos x} \, dx = 2\sqrt{2} \sin\left(\frac{x}{2}\right) + C$$.