Answer
\[ = - \frac{1}{{10}}\,\left( {\frac{1}{{{{\sin }^{10}}x}}} \right) + C\]
Work Step by Step
\[\begin{gathered}
\int_{}^{} {{{\csc }^{10}}x\cot xdx} \hfill \\
\hfill \\
use\,\,\csc t = \frac{1}{{\sin t}}{\text{ and }}\cot t = \frac{{\cos t}}{{\sin t}} \hfill \\
\hfill \\
= \int_{}^{} {\frac{1}{{{{\sin }^{10}}x}} \cdot \frac{{\cos x}}{{\sin x}}dx} \hfill \\
\hfill \\
mutliply \hfill \\
\hfill \\
rewrite\,\,\int_{}^{} {\frac{{\cos x}}{{{{\sin }^{11}}x}}} \,dx \hfill \\
\hfill \\
use\,\,\,\sin x = u\,\,\,\,then\,\,\,\cos xdx = du \hfill \\
\hfill \\
= \int_{}^{} {\frac{{du}}{{{u^{11}}}} = \int_{}^{} {{u^{ - 11}}du} } \hfill \\
\hfill \\
integrate \hfill \\
\hfill \\
= \frac{{{u^{ - 11 + 1}}}}{{ - 10}} + C \hfill \\
\hfill \\
substituting\,\,back\,\,u\, = \,\sin \,x \hfill \\
\hfill \\
= - \frac{1}{{10}}\,\left( {\frac{1}{{{{\sin }^{10}}x}}} \right) + C \hfill \\
\end{gathered} \]