How To Integrate sin^3(x)

 

\begin{array}{l} \smallint si{n^3}x{\rm{d}}x\\\\ \smallint {\sin ^2}x\sin x{\rm{d}}x\\ \smallint \left( {1 - {{\cos }^2}x} \right)\sin x{\rm{d}}x\\ \;\;By\;Substitution\\ \;Suppose \Rightarrow z = \cos x\\ \;{\rm{d}}z = - \sin x{\rm{d}}x\\\\ - \smallint \left( {1 - {z^2}} \right){\rm{d}}z\\ \; \Rightarrow - \left( {z - \frac{{{z^3}}}{3}} \right)\\ \Rightarrow - \left( {\cos x - \frac{{{{\cos }^3}x}}{3}} \right) + c \end{array}