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Question

1sin x-2 cos x2 sin x+cos x dx

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Solution

Let I= 1sin x-2 cos x 2 sin x+cos xdxDividing numerator and denominator by cos2 xI= sec2 x sin x-2 cos xcos x×2 sin x+cos xcos xdx = sec2 x tan x-2 2 tan x+1dxLet tan x=tsec2 x dx=dtI= dtt-2 2t+1 = dt2t2+t-4t-2 = dt2t2-3t-2 =12 dtt2-32t-1 =12 dtt2-32t+342-342-1 =12 dtt-342-916-1 =12 dtt-342-542 =12×12×54 log t-34-54t-34+54+C =15 ln t-2t+12+C =15ln t-222t+1+C =15ln t-22t+1+15 ln 2+C =15 ln t-22t+1+C' where C'=C+15ln 2 =15 ln tan x-22 tan x+1+C

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