Question

高等数学求积分

Answer 1

积化和差公式：sinx*siny=(1/2)[cos(x-y)-cos(x+y)]
sinx*cosy=(1/2)[sin(x-y)+sin(x+y)]
sinx * sin2x * sin3x = (sinx*sin2x) * sin3x
= (1/2)[cos(x-2x)-cos(x+2x)] * sin3x
= (1/2)(cosx-cos3x) * sin3x
= (1/2)(sin3x*cosx) - (1/2)(sin3x*cos3x)
= (1/2)*(1/2)[sin(3x-x)+sin(3x+x)] - (1/2)(sin3x*cos3x)
= (1/4)(sin2x+sin4x) - (1/2)(sin3x*cos3x)
= (1/4)sin2x + (1/4)sin4x - (1/2)(sin3x*cos3x)
∫ (sinx * sin2x * sin3x) dx
= (1/4)∫ sin2x dx + (1/4)∫ sin4x dx - (1/2)∫ sin3x*cos3x dx
= (1/4)(1/2)∫ sin2x d(2x) + (1/4)²∫ sin4x d(4x) - (1/2)(1/3)∫ sin3x*cos3x d(3x)
= (1/8)(-cos2x) + (1/16)(-cos4x) - (1/6)∫ sin3x d(sin3x)
= (-1/8)cos2x - (1/16)cos4x - (1/12)sin²3x + c
= (-1/8)cos3x - (1/16)cos4x + (1/24)cos6x + c