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Question

1sin x-3 cos x dx

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Solution

Let I= dxsin x-3 cos xPutting sin x=2 tan x21+tan2 x2 and cos x=1-tan2 x21+tan2 x2I = 12 tan x21+tan2 x2-31-tan2 x21+tan2 x2dx = 1+tan2 x2 2 tan x2-3+3tan2 x2dx =sec2 x23tan2 x2+2 tan x2-3dx

Let tan x2=t12 sec2 x2dx=dtsec2 x2dx=2dt I=2 dt3t2+2t-3=23 dtt2+23t-1=2dtt2+23t+132-1-132=2 dtt+132-232=22×23log t+13-23t+13+23+C
=32log tanx2-13tanx2+33+C=32log 3 tanx2-13 tanx2+3+C

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