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一先化简再积分 ①直接利用三角公式化简: sinx^8=(1-cos2x)^4/16 =(1+cos2x^2-2cos2x)^2/16 =(3-4cos2x+cos4x)^2/64 =(9+16cos2x^2+cos4x^2+6cos4x-24cos2x-8cos2xcos4x)/64 =(9+8+8cos4x+1/2+cos8x/2+6cos4x-24cos2x-4cos6x-4cos2x)/64 =35/128-7/16cos2x+7/32cos4x-cos6x/16+cos8x/128 ②利用公式sinx=j[Exp(-jx)-Exp(jx)]/2: sinx^8 =[Exp(-jx)-Exp(jx)]/2^8 =1/256∑C(8,n)(-)^nExp[j(2n-8)x] =1/256[2cos8x-16cos6x+56cos4x-112cos2x+70] 二或者用递推,令:F[n]=∫sinx^(2n)dx F[n]=-∫sinx^(2n-1)dcosx =-cosxsinx^(2n-1)+∫cosxd[sinx^(2n-1)] =-cosxsinx^(2n-1)+(2n-1)∫cosx^2 sinx^(2n-2)dx =-cosxsinx^(2n-1)+ (2n-1)(F[n-1]-F[n]) F[n]={-cosxsinx^(2n-1)+(2n-1)F[n-1]}/(2n) F[0]=x