${e^x}\left( {\cfrac{{1 + \sin x}}{{1 + \cos x}}} \right)$

: Let$I = \int {\cfrac{{{e^x}\left( {1 + \sin x} \right)}}{{1 + \cos x}}dx}$

$\Rightarrow$ $I = \int {{e^x}\left[ {\cfrac{{1 + 2\sin \cfrac{x}{2}\cos \cfrac{x}{2}}}{{2{{\cos }^2}\cfrac{x}{2}}}} \right]} = \int {{e^x}\left[ {\cfrac{1}{2}{{\sec }^2}\cfrac{x}{2} + \tan \cfrac{x}{2}} \right]} dx$

$\Rightarrow$ $I = \int {{e^x}\left[ {\tan \cfrac{x}{2} + \cfrac{1}{2}{{\sec }^2}\cfrac{x}{2}} \right]} dx$

$= \int {{e^x}\left[ {\tan \cfrac{x}{2} + \left( {\cfrac{d}{{dx}}\left( {\tan \cfrac{x}{2}} \right)} \right)} \right]} dx$

$= \int {{e^x} \cdot \tan \left( {\cfrac{x}{2}} \right) + C}$