$\underset{0}{\overset{2\pi }{\int }}x{\sin }^{6}x{\cos }^{5}x dx=$

$$\begin{gathered} \begin{array}{l}I=\underset{0}{\overset{2\pi }{\int }}x{\sin }^{6}x.{\cos }^{5}xdx\\ =2\pi \underset{0}{\overset{\pi }{\int }}{\sin }^{6}x{\cos }^{5}xdx\\ \left(\because \underset{0}{\overset{a}{\int }}xf\left(x\right)dx=\frac{a}{2}\underset{0}{\overset{a}{\int }}f\left(x\right)dx if f(a-x)=f\left(x\right) \\ =\frac{a}{2}.2\underset{0}{\overset{a/2}{\int }}f\left(x\right)dx \right)\\ here f\left(x\right)={\sin }^{6}x{\cos }^{5}x\\ f(2\pi -x)={\sin }^6(2\pi -x){\cos }^5(2\pi -x)\\ ={\sin }^{6}x{\cos }^{5}x=f\left(x\right)\\ I=2\pi \underset{0}{\overset{\pi }{\int }}{\sin }^{6}x{\cos }^{5}xdx\\ =2\pi \left(0\right)=0\\ \left(\because \underset{0}{\overset{a}{\int }}f\left(x\right)dx=0 if f(a-x)=-f\left(x\right) here f(\pi -x)={\sin }^6(\pi -x){\cos }^5(\pi -x)\right)\\ ={\sin }^{6}x-(-{\cos }^{5}x)=-f\left(x\right)\end{array} \end{gathered}$$