Proprietatea lui King:
[tex]\displaystyle \int_{a}^{b}f(x)\, dx = \int_{a}^b f(a+b-x)\, dx[/tex]
Rezolvare:
[tex]\displaystyle I =\int_{0}^{\pi}\sin x\cdot \cos x\cdot \cos 2x\cdot \cos 4x\cdot ...\cdot \cos 2^{n-1}x\, dx\\ \\ I = \int_{0}^{\pi}\sin (0+\pi-x)\cdot \cos (0+\pi-x)\cdot \cos \big[2(0+\pi-x)\big]\cdot...\cdot \cos \big[2^{n-1}(0+\pi-x)\big]\, dx\\ \\ I = \int_{0}^{\pi}\sin x\cdot (-\cos x)\cdot \cos(2\pi-2x)\cdot \cos(4\pi-4x)\cdot ...\cdot \cos(2^{n-1}\pi-2^{n-1}x)\, dx\\ \\ I = \int_{0}^{\pi}\sin x\cdot (-\cos x)\cdot \cos 2x\cdot \cos 4x\cdot ...\cdot \cos 2^{n-1}x\, dx\\ \\ \displaystyle I = -\int_{0}^{\pi}\sin x\cdot \cos x\cdot \cos 2x\cdot \cos 4x\cdot ...\cdot \cos 2^{n-1}x\, dx\\ \\ I = -I\\ \\ 2I = 0\\ \\ \Rightarrow \boxed{I = 0}[/tex]
