求数学积分∫(sinx*sin2x*sin3x)dx

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求数学积分∫(sinx*sin2x*sin3x)dx

求数学积分∫(sinx*sin2x*sin3x)dx
求数学积分∫(sinx*sin2x*sin3x)dx

求数学积分∫(sinx*sin2x*sin3x)dx
sinxsin3xsin2x
=-1/2(cos4x-cos2x)sin2x
=1/2(sin2xcos4x-sin2xcos2x)
=1/2*[1/2(sin6x-sin2x)-1/2sin4x]
=1/4*sin6x-1/4*sin2x-1/4*sin4x
原式=1/4∫sin6xdx-1/4∫sin2xdx-1/4∫sin4xdx
=-cos6x/24+cos2x/8+cos4x/16+C

sinxsin3xsin2x
=-1/2(cos4x-cos2x)sin2x
=-1/2(sin2xcos4x-sin2xcos2x)
=-1/2*[1/2(sin6x-sin2x)-1/2sin4x]
=-1/4*sin6x+1/4*sin2x+1/4*sin4x
原式=-1/4∫sin6xdx+1/4∫sin4xdx+1/4∫sin2xdx
=1/24*cos6x-1/16*cos4x-1/8*cos2x+C

即\x-a=rcosθ,y=rsinθ
因为sin2θ+cos2θ=1
所以(x-a)2+y2=r2
所以这是圆心在(a,0),半径是r的圆41151