# Chapter 7 - Integration Techniques - 7.3 Trigonometric Integrals - 7.3 Exercises: 40

$= - \frac{1}{{10}}\,\left( {\frac{1}{{{{\sin }^{10}}x}}} \right) + C$

#### Work Step by Step

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

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