Question

Evaluate ${\int }_0^{\pi }\frac{x^2\sin 2x\cdot \sin \left(\frac{\pi }{2}\cos x\right)dx}{(2x-\pi )}$.

Answer 1

$I={\int }_0^{\pi }\frac{x^{2}sin\left(2x\right)sin\left(\frac{\pi }{2}cosx\right)}{(2x-\pi )}dx$ ----(1)

applying king property

$I={\int }_0^{\pi }\frac{(\pi -x{)}^{2}sin(2x\left)sin\right(\frac{\pi }{2}cosx)}{(2\pi -x)}dx$ ----(2)

adding (1) & (2)

$27=0$

$\Rightarrow$ I=0