How To Integrate 1+sin x /1+cos x

 

\begin{array}{l} \smallint \frac{{{\bf{1}} + {\bf{sin}}x}}{{{\bf{1}} + {\bf{cos}}x}}{\rm{d}}x\\\\ \smallint \frac{{1 + \sin x}}{{1 + \cos x}}\;\; * \;\frac{{1 - cosx}}{{1 - cosx}}{\rm{d}}x\\ \smallint \frac{{\left( {1 + \sin x} \right)\left( {1 - \cos x} \right)}}{{1 - {{\cos }^2}x}}{\rm{d}}x\\ \smallint \frac{{1 + \sin x - \cos x - \sin x\cos x}}{{{{\sin }^2}x}}{\rm{d}}x\\ \smallint \frac{1}{{{{\sin }^2}x}} + \frac{{\sin x}}{{{{\sin }^2}x}} - \frac{{\cos x}}{{{{\sin }^2}x}} - \frac{{\sin x\cos x}}{{{{\sin }^2}x}}dx\\ \smallint {\csc ^2}x + \csc x - \cot x\csc x - \cot x{\rm{d}}x\\ \Rightarrow - \cot x + \ln \left| \; \right.\csc x - \cot x\left| { + \csc x - {{\ln }^\;}\left| \; \right.\sin x} \right| + C \end{array}