Evaluate ${\int }_0^{\pi } x{\sin }^{7}x{\cos }^{6}xdx.$

Let $I={\int }_0^{\pi } x{\sin }^{7}x{\cos }^{6}xdx$

$\left[{\int }_0^a f\left(x\right)dx={\int }_0^a f(a-x)dx\right]$

$={\int }_0^{\pi } (\pi -x){\sin }^7(\pi -x){\cos }^6(\pi -x)dx$

$I={\int }_0^{\pi } (\pi -x){\sin }^{7}x{\cos }^{6}xdx$

$={\int }_0^{\pi } \pi {\sin }^{7}x{\cos }^{6}xdx-{\int }_0^{\pi } x{\sin }^{7}x{\cos }^{6}xdx$

$I=\pi {\int }_0^{\pi } {\sin }^{7}x{\cos }^{6}xdx-I$

$2I=\pi {\int }_0^{\pi } {\sin }^{7}x{\cos }^{6}xdx;2I=2\pi {\int }_0^{\pi /2} {\sin }^{7}x{\cos }^{6}xdx$

$\left[{\int }_0^{\pi /2} {\sin }^{m}x{\cos }^{n}xdx=\frac{{\displaystyle (m-1)(m-3)(m-5)\dots 2\left(\text{ or }\right) 1(n-1)(n-3)(n-5)\dots 2n\left(\text{ or }\right)1}}{{\displaystyle (m+n)(m+n-2)(m+n-4)(m+n-6)\dots .2\left(\text{ or }\right)1}}\right]$

$\left[Where n=\frac{\pi }{2} \mathrm{is} \mathrm{both} m \& n \mathrm{are} \mathrm{even} = 1 \mathrm{other} \mathrm{wise}\right]$

$2I=2\pi \left[\frac{(7-1)(7-3)(7-5)\cdot (6-1)(6-3)(6-5)}{13(13-2)(13-4)(13-6)(13-8)}1\right]$

$2I=2\pi \left[\frac{6\cdot 4\cdot 2\cdot 5\cdot 3\cdot 1}{13\cdot 11\cdot 9\cdot 7\cdot 5\cdot 3\cdot 1}\right]$

$I=\pi \left[\frac{16}{3003}\right]\Rightarrow I=\frac{16\pi }{3003}$