Answer
\[ = \theta \tan \theta + \ln \left| {\cos \theta } \right| + C\]
Work Step by Step
\[\begin{gathered}
\int_{}^{} {\theta {{\sec }^2}\theta d\theta } \hfill \\
\hfill \\
set\,\,\,the\,\,substitution \hfill \\
\hfill \\
u = \theta \,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\, \to \,\,\,\,\,du = d\theta \hfill \\
dv = {\sec ^2}\theta dt\,\,\, \to \,\,\,\,v = \tan \theta \hfill \\
\hfill \\
use\,\,uv - \int_{}^{} {vdu} \hfill \\
\hfill \\
{\text{replacing the values }}{\text{in the equation}} \hfill \\
\hfill \\
= \theta \tan \theta - \int_{}^{} {\tan \theta \,\,d\theta } \hfill \\
or \hfill \\
= \theta \tan \theta + \int_{}^{} {\frac{{ - \sin \theta }}{{\cos \theta }}} d\theta \hfill \\
integrate \hfill \\
\hfill \\
= \theta \tan \theta + \left( {\ln \left| {\cos \theta } \right|} \right) + C \hfill \\
\hfill \\
simplify \hfill \\
\hfill \\
= \theta \tan \theta + \ln \left| {\cos \theta } \right| + C \hfill \\
\end{gathered} \]
