设 $ninbN^+$, 计算积分 $dps{int_0^{pi/2} cfrac{sin nx}{sin x} d x}.$
解答: (1) 由 $$eex ea 2sin xcdot cfrac{1}{2}&=sin x,\ 2sin xcdot cos 2x&=sin 3x-sin x,\ 2sin xcdot cos 4x&=sin 5x-sin 3x,\ cdots&=cdots,\ 2sin xcdot cos 2nx&=sin (2n+1)x-sin(2n-1)x eea eeex$$ 知 $$ex 2sin xsex{cfrac{1}{2}+sum_{k=1}^n cos 2kx}=sin (2n+1)x. eex$$ 于是 $$ex int_0^{pi/2}cfrac{sin (2n+1)x}{sin x} d x =int_0^{pi/2} sex{1+2sum_{k=1}^n cos 2kx} d x =cfrac{pi}{2}. eex$$ (2) 由 $$eex ea 2sin xcos x&=sin 2x,\ 2sin xcos 3x&=sin 4x-sin 2x,\ 2sin xcos 5x&=sin 6x-sin 4x,\ cdots&=cdots,\ 2sin xcos(2n-1)x&=sin 2nx-sin(2n-2)x eea eeex$$ 知 $$ex 2sin xsum_{k=1}^n cos (2k-1)x=sin 2nx. eex$$ 于是 $$ex int_0^{pi/2} cfrac{sin 2nx}{sin x} d x =2int_0^{pi/2} sum_{k=1}^n cos(2k-1)x d x =2sum_{k=1}^n cfrac{(-1)^{k-1}}{2k-1}. eex$$