$\begin{align}  & \int{\frac{{{x}^{4}}}{{{({{x}^{5}}+6)}^{2}}}dx} \\ & \text{Let }u={{x}^{5}}+6,\text{  }\frac{du}{dx}=5{{x}^{4}},\text{ }dx=\frac{du}{5{{x}^{4}}} \\ & \int{\frac{{{x}^{4}}}{{{({{x}^{5}}+6)}^{2}}}dx}=\int{\frac{{{x}^{4}}}{{{u}^{2}}}(\tfrac{du}{5{{x}^{4}}})}=\frac{1}{5}\int{{{u}^{-2}}du} \\ & \int{\frac{{{x}^{4}}}{{{({{x}^{5}}+6)}^{2}}}}=\frac{1}{5}\left[ \frac{{{u}^{-1}}}{-1} \right]+C=-\frac{1}{5}{{u}^{-1}}+C \\ & \int{\frac{{{x}^{4}}}{{{({{x}^{5}}+6)}^{2}}}=-\frac{1}{5({{x}^{5}}+6)}+C} \\\end{align}$