Answer
\[ = \ln \left| {1 + \tan \,\left( {\frac{x}{2}} \right)} \right| + C\]
Work Step by Step
\[\begin{gathered}
\int_{}^{} {\frac{{dx}}{{1 + \sin x + \cos x}}} \hfill \\
\hfill \\
substitution \hfill \\
\hfill \\
set\,\,x = 2{\tan ^{ - 1}}\,\left( u \right)\,\,\,\,\,\,\,then\,\,\,\,\,\,\,\,du = 2\left( {\frac{{du}}{{1 + {u^2}}}} \right) \hfill \\
and\,\,\cos x = \frac{{1 - {u^2}}}{{1 + {u^2}}}\,\,\,then\,\,\,\,\sin x = \frac{{2u}}{{1 + {u^2}}} \hfill \\
\hfill \\
substitute\,\,for\,dx{\text{ }}\sin x{\text{ and }}\cos x \hfill \\
\hfill \\
\int_{}^{} {\frac{{dx}}{{1 + \sin x + \cos x}}} = \int_{}^{} {\frac{{2\frac{{du}}{{1 + {u^2}}}}}{{1 + \frac{{2u}}{{1 + {u^2}}} + \frac{{1 - {u^2}}}{{1 + {u^2}}}}}} \hfill \\
\hfill \\
simplify \hfill \\
\hfill \\
= \int_{}^{} {\frac{{2du}}{{2 + 2u}}} \hfill \\
\hfill \\
integrating \hfill \\
\hfill \\
= \ln \left| {1 + u} \right| + C \hfill \\
\hfill \\
substitute\,\,for\,\,\,u \hfill \\
\hfill \\
= \ln \left| {1 + \tan \,\left( {\frac{x}{2}} \right)} \right| + C \hfill \\
\end{gathered} \]
