How To Integrate sin^5(x)

 

\begin{array}{l} \smallint {\sin ^5}x{\rm{d}}x\\\\ \smallint {\sin ^4}{\rm{x}}\sin {\rm{xdx}}\\ \smallint {\left( {1 - {{\cos }^2}x} \right)^2}\sin x{\rm{d}}x\\ By{\rm{ }}Substitution\;\\ \;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;Suppose\; \Rightarrow z = cosx\;\;\;\;\;\;\;\;\;\\ \;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;dz = - {\rm{ }}sin{\rm{ }}x{\rm{ }}dx\\ - \smallint {\left( {1 - {z^2}} \right)^2}{\rm{d}}z\\ - \smallint 1 - 2{z^2} + {z^4}{\rm{d}}z\;\\ \Rightarrow - \left( {z - \frac{{2{z^3}}}{3} + \frac{{{z^5}}}{5}} \right)\\ \Rightarrow - \left( {\cos x - \frac{2}{3}{{\cos }^3}x + \frac{{{{\cos }^5}x}}{5}} \right) + C \end{array}