Question

高数 求不定积分 总习题四8

Answer 1

用积化和差公式
∫ sinxsin2xsin3x dx
=(1/2)∫ (cosx-cos3x)sin3x dx
=(1/2)∫ (cosxsin3x-cos3xsin3x) dx
=(1/4)∫ (sin4x+sin2x-sin6x) dx
=-(1/16)cos4x-(1/8)cos2x+(1/24)cos6x+C

Answer 2

∫sinxsin2xsin3xdx
=∫sinx * (1/2)[cos(3x-2x)-cos(2x+3x)]dx
=∫sinx * (1/2)(cosx-cos5x)dx
=(1/2)∫sinxcosxdx - (1/2)∫sinxcos5xdx
=(1/2)∫sinxdsinx - (1/4)∫(sinx6x-sin4x)dx
=(1/2)(sin²/2) - (1/4)(-1/6)cos6x + (1/4)(-1/4)cos4x + C
= (1/4)sin²x + (1/24)cos6x - (1/16)cos4x + C