Answer
\[\frac{7}{6}\]
Work Step by Step
\[\begin{gathered}
\int_0^1 {\left( {x + \sqrt x } \right)} dx \hfill \\
\hfill \\
{\text{ radical property }}\sqrt x = {x^{1/2}} \hfill \\
\hfill \\
\int_0^1 {\left( {x + {x^{1/2}}} \right)} dx \hfill \\
\hfill \\
{\text{Integrate using the rule }}\int {{x^n}} dx = \frac{{{x^{n + 1}}}}{{n + 1}} + C \hfill \\
\hfill \\
\left[ {\frac{{{x^{1 + 1}}}}{{1 + 1}} + \frac{{{x^{1/2 + 1}}}}{{1/2 + 1}}} \right]_0^1 \hfill \\
\hfill \\
\left[ {\frac{{{x^2}}}{2} + \frac{{{x^{3/2}}}}{{3/2}}} \right]_0^1 \hfill \\
\hfill \\
\left[ {\frac{{{x^2}}}{2} + \frac{2}{3}{x^{3/2}}} \right]_0^1 \hfill \\
\hfill \\
{\text{Fundamental Theorem of calculus}} \hfill \\
\hfill \\
\left[ {\frac{{{{\left( 1 \right)}^2}}}{2} + \frac{2}{3}{{\left( 1 \right)}^{3/2}}} \right] - \left[ {\frac{{{{\left( 0 \right)}^2}}}{2} + \frac{2}{3}{{\left( 0 \right)}^{3/2}}} \right] \hfill \\
\hfill \\
{\text{Simplify}} \hfill \\
\hfill \\
\frac{1}{2} + \frac{2}{3} \hfill \\
\hfill \\
\frac{7}{6} \hfill \\
\end{gathered} \]
