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Question
Find the integrals of the function:
`(cos x - sinx)/(1+sin 2x)`
Solution
Let `I = int (cos x - sin x)/ (1 + sin 2x) dx`
`= int (cos x - sin x)/(1 + 2 sinx cosx) dx` .... [∵ sin2x = 2sinx cosx]
`= int (cosx - sin x)/(sin^2x + cos^2x + 2 sin x cos x) dx`
`= int (cos x - sin x)/ (cos x + sinx)^2 dx`
Put cos x + sin x = t
⇒ (- sin x + cos x) dx = dt
∴ `I = intdt/t^2 = t^(-2+1)/(-2 + 1) + C`
`= (-1)/t + C`
`= -1/ (cos x + sin x) + C`
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