Answer
\[ = t\,{e^t}\, - \,{e^t}\, + \,C\]
Work Step by Step
\[\begin{gathered}
\int_{}^{} {t\,{e^t}\,dt} \hfill \\
\hfill \\
set\,\,\,the\,\,substitution \hfill \\
\hfill \\
u = t\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,then\,\,\,\,\,\,\,\,du = dt \hfill \\
du = {e^t}dt\,\,\,\,\,then\,\,\,\,\,\,\,\,\,v = {e^t} \hfill \\
\hfill \\
use\,\,uv - \int_{}^{} {vdu} \hfill \\
\hfill \\
{\text{replacing the values }}{\text{in the equation}} \hfill \\
\hfill \\
t\,{e^t} - \int_{}^{} {{e^t}dt} \hfill \\
\hfill \\
integrate \hfill \\
\hfill \\
= t\,{e^t}\, - \,{e^t}\, + \,C \hfill \\
\end{gathered} \]
