Answer
\[ - \frac{1}{{2\,{{\left( {x + 5} \right)}^4}}} + \frac{2}{{\,{{\left( {x + 5} \right)}^5}}} + C\]
Work Step by Step
\[\begin{gathered}
\int_{}^{} {\frac{{2x}}{{\,{{\left( {x + 5} \right)}^6}}}dx} \hfill \\
Let\,u = x + 5\,\,,\,\,then\,\,x = u - 5 \hfill \\
So\,\,that \hfill \\
dx = du \hfill \\
Then \hfill \\
\int_{}^{} {\frac{{2\,\left( {u - 5} \right)}}{{{u^6}}}du\,} \hfill \\
Write\,\,the\,\,integrand\,\,as \hfill \\
2\int_{}^{} {\,\left( {\frac{u}{{{u^6}}} - \frac{5}{{{u^6}}}} \right)du} \hfill \\
2\int_{}^{} {\,\left( {{u^{ - 5}} - 5{u^{ - 6}}} \right)du} \hfill \\
Integrating \hfill \\
2\,\left( {\frac{{{u^{ - 4}}}}{{ - 4}}} \right) - 10\,\left( {\frac{{{u^{ - 5}}}}{{ - 5}}} \right) + C \hfill \\
- \frac{1}{{2{u^4}}} + \frac{2}{{{u^5}}} + C \hfill \\
Substituting\,\,u = x + 5\,\,gives \hfill \\
- \frac{1}{{2\,{{\left( {x + 5} \right)}^4}}} + \frac{2}{{\,{{\left( {x + 5} \right)}^5}}} + C \hfill \\
\end{gathered} \]