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Question
`int (cos 2x)/(sin x + cos x)^2dx` is equal to ______.
Options
`(- 1)/(sin x + cos x) + "C"`
log |sin x + cos x| + C
log |sin x - cos x| + C
`1/(sin x + cos x)^2`
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Solution
`int (cos 2x)/(sin x + cos x)^2dx` is equal to log |sin x + cos x| + C.
Explanation:
Let `I = (cos 2x)/(sin x + cos x) dx`
`= int (cos^2 x - sin^2 x)/(cos x + sin x)^2 dx`
`= int ((cos x - sin x)(cos x + sin x))/(cos x + sin x)^2 dx`
`= int (cos x - sin x)/(cos x + sin x) dx`
put cos x + sin x = t
⇒ (-sin x + cos x)dx = dt
`= int dt/t = log t + C`
= log |sin x + cos x| + C
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