71.tex

\def\no{71}
\def\theintegral{\(\int\sec^3{u}\,\dif{u}
\,=\,
\frac{1}{2}\sec{u}\,\tan{u}
+\frac{1}{2}\ln\abs{\sec{u}+\tan{u}}+C
\)}

% \def\aroundarrows{\mspace{18mu}}
% \def\andsoarrow{\Rightarrow}
\begin{align*}
% \renewcommand{\baselinestretch}{2}
{I}_{\no}
=&  \int  \sec^3 u  \dif u
\parts*{\sec u}   {\sec u \tan u}
      {\sec^2 u} {\tan u}
\\
=&  \sec u \tan u - \int  \sec u \tan^2 \dif u
\\
=&  \sec u \tan u - \int  \sec u (\sec^2 u - 1) \dif u
\\
=&  \sec u \tan u - \int  \sec^3 u  \dif u
                  + \int  \sec u    \dif u
\\
2 \int  \sec^3 u  \dif u
=& \sec u \tan u + \int  \sec u    \dif u
\\
\int  \sec^3 u  \dif u
=&  \frac{1}{2}\sec u \tan u
  + \frac{1}{2}\int  \sec u  \dif u
\\
=&  \frac{1}{2}\sec u \tan u
  + \frac{1}{2}\ln\abs{\sec u+\tan u} +C
%
\end{align*}
%