$\int \frac{\cos x-\sin x}{\cos x+\sin x}(2+2 \sin 2 x) d x$ is equal to

$\sin 2 x+C$ $\cos 2 x+C$ $\tan 2 x+C$ none of these

$\sin 2 x+C$

Let $I=\int \frac{\cos x-\sin x}{\cos x+\sin x}(2+2 \sin 2 x) d x$ $\Rightarrow I =2 \int \frac{\cos x-\sin x}{\cos x+\sin x} \times(\cos x+\sin x)^2 d x$ $\Rightarrow I =\int\left(\cos ^2 x-\sin ^2 x\right) d x=2 \int \cos 2 x d x=\sin 2 x+C$