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sinx与sin2x

这样

n=1时公式成立; 现在假设对n-1公式成立 那么sinx+sin2x+sin3x+……+sinnx=sinx+sin2x+sin3x+……+sin(n-1)x+sinnx =[sin((n-1)x/2)sin(nx/2)]/sin(x/2)+sinnx =[sin((n-1)x/2)sin(nx/2)+sinnxsin(x/2)]/sin(x/2) =sin(nx/2)[sin((nx/2-x/2)+2cos(nx...

利用积化和差公式-2sin((A+B)/2)*sin((A-B)/2)=cosA-cosB 2sin(x/2)*sinx=cos(x/2)-cos(3x/2) 2sin(x/2)*sin2x=cos(3x/2)-cos(5x/2) ... 2sin(x/2)*sinnx=cos((2n-1)x/2)-cos((2n+1)x/2) 裂项相消 原式就等于cos(x/2)-cos((2n+1)x/2)

如图所示:

∫sinxsin2xdx =-1/2∫(cos3x-cosx)dx =-1/2[1/3sin3x-sinx]+C =-1/6sin3x+1/2sinx+C

sin2x=2sinxcosx

因为cosX- cos3X =cos(2x-x)-cos(2x+x) =cos2xcosx+sin2xsinx -(cos2xcosx-sin2xsinx) =2sin2xsinx

和自变量数列求和有关的公式 sinx+sin2x+sin3x+……+sinnx=[sin(nx/2)sin((n+1)x/2)]/sin(x/2) cosx+cos2x+cos3x+……+cosnx=[cos((n+1)x/2sin(nx/2)]/sin(x/2) tan((n+1)x/2)=(sinx+sin2x+sin3x+……+sinnx)/(cosx+cos2x+cos3x+……+cosnx) sinx+sin3x+...

两角和的正弦公式 sin(x+x) = sinx cosx + cosx sinx = 2 sinx cosx

∫(sinxsin2x)dx =2∫sin²xcosxdx =2∫sin²xdsinx =2sin³x/3+C

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