Question

The value of $\int \frac{\sin x+\cos x}{2-\sin 2x}dx$ is
(where $C$ is constant of integration)

• A. ${\tan }^{-1}(\sin x-\cos x)+C$
• B. ${\tan }^{-1}(\sin x+\cos x)+C$
• C. $2{\tan }^{-1}(\cos x-\sin x)+C$
• D. $2{\tan }^{-1}(\sin x+\cos x)+C$

Answer 1

The correct option is A ${\tan }^{-1}(\sin x-\cos x)+C$

Let
$I=\int \frac{\sin x+\cos x}{2-\sin 2x}dx=\int \frac{\sin x+\cos x}{1+(\sin x-\cos x{)}^2}dx$
put
$\sin x-\cos x=y,(\cos x+\sin x)dx=dy$
$\Rightarrow I=\int \frac{dy}{1+y^2}={\tan }^{-1}y+C\Rightarrow I={\tan }^{-1}(\sin x-\cos x)+C$