77.tex

\def\no{77}
\def\theintegral{\(\int\sec^n{u}\;\dif{u}
\enspace=\enspace%
\tfrac{1}{n-1}\,\tan u\,\sec^{n-2}{u}
\,+\,
\tfrac{n-2}{n-1}\,\int\sec^{n-2}u\;\dif{u}
\)}

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