int (1-cos 3x)/(1-cos x)1cos3x1cosx ?

1 Answer
May 31, 2018

sin2x+4sinx+3x+Csin2x+4sinx+3x+C.

Explanation:

Since, cos3x-1=(4cos^3x-3cosx)-1cos3x1=(4cos3x3cosx)1, we find that,

cos3x-1cos3x1 is a cubic polynomial in cosxcosx, and noting that the sum

of its co-efficients being 0, (cosx-1)0,(cosx1) is a factor.

Thus, cos3x-1=4cos^3x-3cosx-1cos3x1=4cos3x3cosx1,

=ul(4cos^3x-4cos^2x)+ul(4cos^2x-4cosx)+ul(cosx-1),

=4cos^2x(cosx-1)+4cosx(cosx-1)+1(cosx-1),

=(cosx-1)(4cos^2x+4cosx+1)

=(cosx-1){2(1+cos2x)+4cosx+1},

:. (cos3x-1)=(cosx-1)(2cos2x+4cosx+3).

rArr (cos3x-1)/(cosx-1)=(2cos2x+4cosx+3).

:. int(1-cos3x)/(1-cosx)dx=int(cos3x-1)/(cosx-1)dx,

=int(2cos2x+4cosx+3)dx,

=2*(sin2x)/2+4sinx+3x.

rArr int(1-cos3x)/(1-cosx)dx=sin2x+4sinx+3x+C.

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