Answer
\[ - 4{x^{ - 2.5}} + 4\ln \left| x \right| + C\]
Work Step by Step
\[\begin{gathered}
\int_{}^{} {\,\left( {10{x^{ - 3.5}} + 4{x^{ - 1}}} \right)dx} \hfill \\
Use\,\,the\,\,sum\,\,and\,\,difference\,\,rules \hfill \\
\int_{}^{} {10{x^{ - 3.5}}dx + \int_{}^{} {4{x^{ - 1}}dx} } \hfill \\
10\int_{}^{} {{x^{ - 3.5}}dx} + 4\int_{}^{} {{x^{ - 1}}dx} \hfill \\
\,use\,\,the\,\,power\,\,rule\,\,\,and \hfill \\
\int_{}^{} {\frac{1}{x}dx} = \ln \left| x \right| + C \hfill \\
Then \hfill \\
10\,\left( {\frac{{{x^{ - 3.5 + 1}}}}{{ - 3.5 + 1}}} \right) + 4\,\left( {\ln \left| x \right|} \right) + C \hfill \\
10\,\left( {\frac{{{x^{ - 2.5}}}}{{ - 2.5}}} \right) + 4\ln \left| x \right| + C \hfill \\
Simplifying \hfill \\
- 4{x^{ - 2.5}} + 4\ln \left| x \right| + C \hfill \\
\end{gathered} \]
