96.tex

\def\no{96}
\def\theintegral{\(\int{ue^{au}}\;\dif{u}%
\enspace=\enspace%
\frac{1}{a^2}{\left(au-1\right)}e^{au}\;+\;C\)}

\begin{align*}
%
{I}_{\no}
=&  \int u e^{au} \dif u
\parts*
{u}{1}
{e^{au}}{\frac{e^{au}}{a}}
\\
=&  \frac{1}{a}ue^{au}
  - \frac{1}{a} \int e^{au} \dif u
\\
=&  \frac{1}{a}ue^{au}
  - \frac{1}{a^2}e^{au} + C
\\
=&  \left(
      \frac{1}{a}u
    - \frac{1}{a^2}
    \right) e^{au} + C
\nb{\simplify}
%
%
=&  \frac{1}{a^2}\left(au-1\right)e^{au} + C
%
\end{align*}