Mathematics - Definite Integration Question with Solution | TestHub

Let $I={\int }_0^{\pi }x{\sin }^{5}x\cdot {\cos }^{6}xdx={\int }_0^{\pi }(\pi -x)\cdot {\sin }^5(\pi -x)\cdot {\cos }^6(\pi -x)dx$

$$={\int }_0^{\pi }(\pi -x){\sin }^{5}x\cdot {\cos }^{6}xdx$$

$={\int }_0^{\pi }\pi {\sin }^{5}x\cdot {\cos }^{6}xdx-{\int }_0^{\pi }x{\sin }^{5}x{\cos }^{6}xdx$ $\Rightarrow 2I=\pi \cdot 2{\int }_0^{\pi /2}{\sin }^{5}x\cdot {\cos }^{6}xdx\Rightarrow I=\pi \left[\frac{4}{11}\cdot \frac{2}{9}\cdot \frac{1}{7}\right]=\frac{8\pi }{693}$.