78.tex

\def\no{78}
\def\theintegral{\(\int\csc^n{u}\;\dif{u}
\enspace=\enspace%
\tfrac{-1}{n-1}\,\cot u\,\csc^{n-2}{u}
\,+\,
\tfrac{n-2}{n-1}\,\int\csc^{n-2}u\;\dif{u}
\)}

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