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Question
Integrate the functions:
`1/(1 - tan x)`
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Solution
Let `I = int 1/ (1 - tan x)dx = int 1/ (1 - sin x/ cos x) dx`
`= int cos x/ (cos x - sin x) dx = 1/2 int (2 cos x)/ (cos x - sin x) dx`
`1/2 int ((cos x - sin x) - (-sin x - cos x))/(cos x - sin x)`
`1/2 int 1 dx - 1/2 int (-sin x - cos x)/ (cos x - sin x) dx`
`x/2 - 1/2 int (-sin x - cos x)/ (cos x - sin x) dx + C_1`
`I = x/2 - 1/2 I_1 + C_1` ....(i)
Where, `I_1 = int (-sinx - cos x)/(cos x - sin x) dx`
Put cos x - sin x = t
⇒ (-sin x - cos x) dx = dt
`I_1 = int dt/t = log |t| + C_2`
= log | cos x - sin x| + C2 ...(ii)
From (i) and (ii), we get
⇒ `I = x/2 - 1/2 log |cos x - sin x| + C`
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