Answer
\[\frac{{10}}{3}{z^{3/2}} + \sqrt 2 z + C\]
Work Step by Step
\[\begin{gathered}
\int_{}^{} {\,\left( {5\sqrt z + \sqrt 2 } \right)dz} \hfill \\
write\,\,\sqrt z \,\,as\,\,{z^{1/2}} \hfill \\
\int_{}^{} {\,\left( {5{z^{1/2}} + \sqrt 2 } \right)} dz \hfill \\
Extending\,\,using\,\,the\,\,sum\,\,and \hfill \\
difference\,\,rules \hfill \\
\int_{}^{} {5{z^{1/2}}\,dz} \int_{}^{} {\sqrt 2 dz} \hfill \\
Use\,\,the\,\,power\,\,rule \hfill \\
\int_{}^{} {{x^n}dx} = \frac{{{x^{n + 1}}}}{{n + 1}} + C \hfill \\
5\,\left( {\frac{{{z^{3/2}}}}{{3/2}}} \right) + \sqrt 2 z + C \hfill \\
Simplifying \hfill \\
\frac{{10}}{3}{z^{3/2}} + \sqrt 2 z + C \hfill \\
\end{gathered} \]
