Answer
\[\ln 5\]
Work Step by Step
\[\begin{align}
& \int_{0}^{4}{\int_{\sqrt{x}}^{2}{\frac{3}{2+{{y}^{3}}}}dydx} \\
& \text{Switch the order of integration using the region shown below} \\
& \int_{0}^{4}{\int_{\sqrt{x}}^{2}{\frac{3}{2+{{y}^{3}}}}dydx}=\int_{0}^{2}{\int_{0}^{{{y}^{2}}}{\frac{3}{2+{{y}^{3}}}}dxdy} \\
& \text{Integrating} \\
& =\int_{0}^{2}{\left[ \frac{3x}{2+{{y}^{3}}} \right]_{0}^{{{y}^{2}}}dy} \\
& =\int_{0}^{2}{\frac{3{{y}^{2}}}{2+{{y}^{3}}}dy} \\
& =\left[ \ln \left| 2+{{y}^{3}} \right| \right]_{0}^{2} \\
& =\ln \left| 2+{{\left( 2 \right)}^{3}} \right|-\ln \left| 2+{{\left( 0 \right)}^{3}} \right| \\
& =\ln \left( 10 \right)-\ln \left( 2 \right) \\
& =\ln 5 \\
\end{align}\]
