Answer
\[\frac{\ln 33}{10}\]
Work Step by Step
\[\begin{align}
& \int_{0}^{4}{\int_{\sqrt{x}}^{2}{\frac{x}{{{y}^{5}}+1}}dydx} \\
& y=\sqrt{x}\to x={{y}^{2}} \\
& \text{Using the graph to switch the order of integration} \\
& R=\left\{ \left( x,y \right):0\le x\le {{y}^{2}},\text{ 0}\le y\le 2\text{ } \right\} \\
& \text{Then}, \\
& \int_{0}^{4}{\int_{\sqrt{x}}^{2}{\frac{x}{{{y}^{5}}+1}}dydx}=\int_{0}^{2}{\int_{0}^{{{y}^{2}}}{\frac{x}{{{y}^{5}}+1}}dxdy} \\
& \text{Integrating} \\
& =\frac{1}{2}\int_{0}^{2}{\left[ \frac{{{x}^{2}}}{{{y}^{5}}+1} \right]_{0}^{{{y}^{2}}}dy} \\
& =\frac{1}{2}\int_{0}^{2}{\frac{{{y}^{4}}}{{{y}^{5}}+1}dy} \\
& =\frac{1}{10}\int_{0}^{2}{\frac{5{{y}^{4}}}{{{y}^{5}}+1}dy} \\
& =\frac{1}{10}\left[ \ln \left( {{y}^{5}}+1 \right) \right]_{0}^{2} \\
& =\frac{1}{10}\left[ \ln \left( {{2}^{5}}+1 \right)-\ln \left( {{0}^{5}}+1 \right) \right] \\
& =\frac{1}{10}\left[ \ln \left( 33 \right)-\ln \left( 1 \right) \right] \\
& =\frac{\ln 33}{10} \\
\end{align}\]
