Answer
\[ = \frac{1}{9}\,{\left( {{x^{\frac{3}{2}}} + 8} \right)^6} + C\]
Work Step by Step
\[\begin{gathered}
\int_{}^{} {\,{{\left( {{x^{\frac{3}{2}}} + 8} \right)}^5}\sqrt x } dx \hfill \\
\hfill \\
u = {x^{\frac{3}{2}}} + 8\,\,\,\,\,\,then\,\,\,\,du = \frac{3}{2}\sqrt x \,\,dx \hfill \\
\hfill \\
apply\,\,the\,\,\,substitution \hfill \\
\hfill \\
= \frac{2}{3}\int_{}^{} {{u^5}du} \hfill \\
\hfill \\
integrate \hfill \\
\hfill \\
= \frac{1}{9}{u^6} + C \hfill \\
\hfill \\
replace\,\,u\,\,with\,\,u = {x^{\frac{3}{2}}} + 8 \hfill \\
\hfill \\
= \frac{1}{9}\,{\left( {{x^{\frac{3}{2}}} + 8} \right)^6} + C \hfill \\
\end{gathered} \]
