Question

$ \int{\frac{\cos x-\sin x}{1+2\sin x\cos x}}dx $ is equal to

• A. $ -\frac{1}{\cos x-\sin x}+C $
• B. $ \frac{\cos x+\sin x}{\cos x-\sin x}+C $
• C. $ -\frac{1}{\sin x+\cos x}+C $
• D. $ \frac{1}{\sin x+\cos x}+C $
• E. $ \tan x+\sec x+C$

Answer 1

Answer: (C)

Let $ I=\int{\frac{\cos x-\sin x}{{{(\cos x+\sin x)}^{2}}}}dx $
Put $ \cos x+\sin x=t $
$ \Rightarrow $ $ (\cos x-\sin x)dx=dt $
$ \therefore $ $ I=\int{\frac{1}{{{t}^{2}}}}dt $
$ I=-\frac{1}{t}+c=-\frac{1}{\sin x+\cos x}+c $