Question

Evaluate $\int \frac{\cos x+\sin x}{\cos x-\sin x}dx$
(where $C$ is constant of integration)

• A. $\frac{\ln |{\sec }^{2}2x+\sec 2x\cdot \tan 2x|}{2}+C$
• B. $\frac{\ln |{\tan }^{2}2x+\sec 2x\cdot \tan 2x|}{2}+C$
• C. $\frac{\ln |\sec 2x\cdot \tan 2x|}{2}+C$
• D. $\frac{\ln |1-{\tan }^{2}2x|}{2}+C$

Answer 1

The correct option is A $\frac{\ln |{\sec }^{2}2x+\sec 2x\cdot \tan 2x|}{2}+C$

Let $I=\int \frac{\cos x+\sin x}{\cos x-\sin x}dx$
$\Rightarrow I=\int \frac{(\cos x+\sin x)\cdot (\cos x+\sin x)}{{\cos }^{2}x-{\sin }^{2}x}dx$
$\Rightarrow I=\int \frac{1+\sin 2x}{\cos 2x}dx$
$\Rightarrow I=\int \sec 2xdx+\int \tan 2xdx$
$\Rightarrow I=\frac{\ln |\sec 2x+\tan 2x|}{2}+\frac{\ln |\sec 2x|}{2}+C$
$\therefore I=\frac{\ln |{\sec }^{2}2x+\sec 2x\cdot \tan 2x|}{2}+C$