Answer
\[ = \frac{{\,{{\left( {{x^2} + 1} \right)}^5}}}{5} + C\]
Work Step by Step
\[\begin{gathered}
\int_{}^{} {2x\,{{\left( {{x^2} + 1} \right)}^4}dx} \hfill \\
\hfill \\
set\,\,\,u = {x^2} + 1\,\,\,then\,\,\,\,du = 2xdx \hfill \\
\hfill \\
therefore \hfill \\
\hfill \\
= \int_{}^{} {{u^4}du} \hfill \\
\hfill \\
{\text{integrating}} \hfill \\
\hfill \\
= \frac{{{u^5}}}{5} + C \hfill \\
\hfill \\
replace\,\,u\,\,with\,u = {x^2} + 1 \hfill \\
\hfill \\
= \frac{{\,{{\left( {{x^2} + 1} \right)}^5}}}{5} + C \hfill \\
\hfill \\
\end{gathered} \]