How To Integrate sin^3(x)cos^2(x)

 

\begin{array}{l} \smallint si{n^3}xco{s^2}x{\rm{d}}x\\\\ \smallint {\bf{si}}{{\bf{n}}^{\bf{2}}}x{\bf{co}}{{\bf{s}}^{\bf{2}}}x{\bf{sin}}x{\rm{d}}x\\ \smallint \left( {1 - {{\cos }^2}x} \right){\cos ^2}x\;sinx\;{\rm{d}}x\\ By\;Substitution\;\;\;\;\;\;\\ \;\;\;\;\;Suppose \Rightarrow z = cosx\\ dz = - \;\sin xdx\\ - \smallint \left( {{\bf{1}} - {z^{\bf{2}}}} \right){z^{\bf{2}}}{\rm{d}}z\\ \smallint {z^{\bf{4}}} - {z^{\bf{2}}}{\rm{d}}z\\ \Rightarrow \frac{{{z^{\bf{5}}}}}{{\bf{5}}} - \frac{{{z^{\bf{3}}}}}{{\bf{3}}} + c\\ \Rightarrow \frac{{{\bf{co}}{{\bf{s}}^{\bf{5}}}x}}{{\bf{5}}} - \frac{{{\bf{co}}{{\bf{s}}^{\bf{3}}}x}}{{\bf{3}}} + c \end{array}