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$\int \frac{\cos 4 x - 1}{\cot x - \tan x}$ dx is equal to

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a

−12cos⁡4x+C

b

−14cos⁡4x+C

c

−12sin⁡2x+C

d

none of these

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detailed solution

Correct option is D

We can write I=∫−2sin2⁡2xcos⁡xsin⁡xcos2⁡x−sin2⁡xdx=−∫sin3⁡2xcos⁡2xdx=−∫sin3⁡2x1−sin2⁡2xcos⁡2xdxPut sin⁡2x=t to obtain I=−12∫t31−t2dt=12∫t1−t2−t1−t2dt=14t2+14log⁡1−t2+C=14sin2⁡2x+12log⁡|cos⁡2x|+C.

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