Let $I = \int {\frac{{dx}}{{1 + \cos x}}} = \int {\frac{{dx}}{{1 + 2{{\cos }^2}\frac{x}{2} – 1}}} $

$ = \frac{1}{2}\int {\frac{1}{{{{\cos }^2}\frac{x}{2}}}} dx = \frac{1}{2}\int {{{\sec }^2}} \frac{x}{2}dx$

$ = \frac{1}{2} \cdot \tan \frac{x}{2} \cdot 2 + C = \tan \frac{x}{2} + C$

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