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Question

Let f(x)=sin3x+sin3(x+2π3)+sin3(x+4π3) then the primitive of f(x) w.r.t. x is

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Solution

f(x)=sin3x+sin3(x+2π3)+sin3(x+4π3)=3sinx4sin3x4+3sin(x+2π3)4sin(3x+2π)4+3sin(x+4π3)4sin(3x+4π)4=14(3sinxsin3x+3sin(x+ππ3)sin3x+3sin(x+π+π3)sin3x)=14(3sinx3sin(xπ3)3sin(x+π3)3sin3x)=14(3sinx3(sinxcosπ3sinπ3cosx+sinxcosπ3+sinπ3cosx)3sin3x)=14(3sinx3×2sinxcosπ33sin3x)=3sin3x4
3sin3x4dx=3cos3x4×3+C=cos3x4+C

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