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