The integral $I=\int \frac{2\sin x}{\left(3+\sin 2x\right)}dx$ simplifies to (where, $C$ is the constant of integration)

• A

  $\ln \left|\frac{2+\sin x-\cos x}{2-\sin x+\cos x}\right|-{\tan }^{-1}\left(\sin x+\cos x\right)+C$
• B

  $\ln (\sin x)+\sin 2x+C$
• C

  $\sin \left(2x\right)-\ln (\cos x)+C$
• Correct answer
  D

  $\frac{1}{4}\ln \left|\frac{2+\sin x-\cos x}{2-\sin x+\cos x}\right|-\frac{1}{\sqrt{2}}{\tan }^{-1}\left(\frac{\sin x+\cos x}{\sqrt{2}}\right)+C$

Hence, the correct answer is option: 'D'.