${\int }_0^{2\mathrm{\pi }} {\mathrm{xsin}}^4{\mathrm{xcos}}^6\mathrm{xdx}=$

$$\begin{gathered} \mathrm{Let} \mathrm{I}={\int }_0^{2\mathrm{\pi }}\mathrm{x} {\sin }^4{\mathrm{xcos}}^6\mathrm{x}d\mathrm{x} \\ \mathrm{Apply} \mathrm{the} \mathrm{property} \boxed{{\int }_0^{\mathrm{a}}\mathrm{f}\left(\mathrm{x}\right)d\mathrm{x}={\int }_0^{\mathrm{a}}\mathrm{f}(\mathrm{a}-\mathrm{x})d\mathrm{x}} \\ Then given integral becomes \\ \mathrm{I}={\int }_0^{2\mathrm{\pi }} (2\mathrm{\pi }-\mathrm{x}){\sin }^4(2\mathrm{\pi }-\mathrm{x}){\cos }^6(2\mathrm{\pi }-\mathrm{x})\mathrm{dx} \\ \mathrm{I}={\int }_0^{2\mathrm{\pi }} (2\mathrm{\pi }-\mathrm{x}){\sin }^4{\mathrm{xcos}}^6\mathrm{x} \mathrm{dx} \\ \mathrm{I}={\int }_0^{2\mathrm{\pi }}2{\mathrm{\pi sin}}^4{\mathrm{xcos}}^6\mathrm{x} \mathrm{dx}-{\int }_0^{2\mathrm{\pi }}\mathrm{x} {\sin }^4{\mathrm{xcos}}^6\mathrm{x}d\mathrm{x} \\ \Rightarrow I=2\mathrm{\pi }{\int }_0^{2\mathrm{\pi }} {\sin }^4{\mathrm{xcos}}^6\mathrm{xdx}-I \\ \Rightarrow 2\mathrm{I}=2\mathrm{\pi }{\int }_0^{2\mathrm{\pi }} {\sin }^4{\mathrm{xcos}}^6\mathrm{xdx} \\ \Rightarrow I=\mathrm{\pi }{\int }_0^{2\mathrm{\pi }} {\sin }^4{\mathrm{xcos}}^6\mathrm{xdx} \\ \Rightarrow I= 4\pi {\int }_0^{\raisebox{1ex}{$\mathrm{\pi }$}\!\left/ \!\raisebox{-1ex}{$2$}\right.} {\sin }^4{\mathrm{xcos}}^6\mathrm{xdx} where m=4,n=6 \\ I=4\pi \left[\frac{3}{10}.\frac{1}{8}.\frac{5}{6}.\frac{3}{4}.\frac{1}{2}.\frac{\pi }{2}\right] \\ \Rightarrow I =\frac{3{\mathrm{\pi }}^2}{128} \end{gathered}$$