Question

\[\int\frac{x + \sin x}{1 + \cos x} \text{ dx }\]

Answer 1

\[\int\left( \frac{x + \sin x}{1 + \cos x} \right)dx\]
\[ = \int\left[ \frac{x}{1 + \cos x} + \frac{\sin x}{1 + \cos x} \right]dx\]
\[ = \int\left[ \frac{x}{2 \cos^2 \frac{x}{2}} + \frac{2 \sin \frac{x}{2} \cos \frac{x}{2}}{2 \cos^2 \frac{x}{2}} \right]dx\]
\[ = \frac{1}{2}\int x_I \cdot \sec^2_{II} \frac{x}{2}dx + \int\tan \frac{x}{2}dx\]
\[ = \frac{1}{2}\left[ x \cdot \frac{\tan \left( \frac{x}{2} \right)}{\frac{1}{2}} - \int1 \times 2 \tan \left( \frac{x}{2} \right)dx \right] + \frac{\text{ log }\left| sec \frac{x}{2} \right|}{\frac{1}{2}} + C\]
\[ = x \tan \left( \frac{x}{2} \right) - \frac{\text{ log} \left| \sec \frac{x}{2} \right|}{\frac{1}{2}} + \text{ log} \frac{\left| \sec \frac{x}{2} \right|}{\frac{1}{2}} + C\]
\[ = x \tan \left( \frac{x}{2} \right) + C\]