Answer
\[\frac{{{e^{{x^2}}}}}{2} + C\]
Work Step by Step
\[\begin{gathered}
\int_{}^{} {x{e^{{x^2}}}dx} \hfill \\
\hfill \\
set\,\,the\,\,substitution \hfill \\
\hfill \\
u = {x^2}\,\,\,\,\,\,then\,\,\,\,\,du = 2xdx \hfill \\
\hfill \\
\frac{1}{2}\int_{}^{} {{e^u}du} \hfill \\
\hfill \\
integrate\,\, \hfill \\
\hfill \\
= \frac{{{e^u}}}{2} + C \hfill \\
\hfill \\
replace\,\,u\,\,with\,\,\,u = {x^2} \hfill \\
\hfill \\
\frac{{{e^{{x^2}}}}}{2} + C \hfill \\
\end{gathered} \]