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. $\frac{1}{\sin x+\cos x}+C$

Answer 1

Answer: (C)

$-\frac{1}{\sin x+\cos x}+C$

Answer 2

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