Answer
\[ = \frac{{{{\sin }^6}x}}{6} + \frac{{3{{\sin }^4}x}}{4} - \frac{{{{\sin }^2}x}}{2} + C\]
Work Step by Step
\[\begin{gathered}
\int_{}^{} {\left( {{{\sin }^5}x + 3{{\sin }^3}x - \sin x} \right)\cos x\,dx} \hfill \\
\hfill \\
distribute \hfill \\
\hfill \\
= \int_{}^{} {{{\sin }^5}x\cos xdx + 3} \int_{}^{} {{{\sin }^3}x\cos xdx} - \int_{}^{} {\sin x\cos xdx} \hfill \\
\hfill \\
integrate\,, \hfill \\
\hfill \\
= \frac{{{{\sin }^6}x}}{6} + 3\,\left( {\frac{{{{\sin }^4}x}}{4}} \right) - \frac{{{{\sin }^2}x}}{2} + C \hfill \\
\hfill \\
simplify \hfill \\
\hfill \\
= \frac{{{{\sin }^6}x}}{6} + \frac{{3{{\sin }^4}x}}{4} - \frac{{{{\sin }^2}x}}{2} + C \hfill \\
\end{gathered} \]
