The value of $$\sqrt{2} \int \frac{\sin x\, dx}{\sin\left(x - \frac{\pi}{4}\right)}$$ is

A

$$x + \log\left|\cos\left(x - \frac{\pi}{4}\right)\right| + c$$

B

$$x - \log\left|\sin\left(x - \frac{\pi}{4}\right)\right| + c$$

C

$$x + \log\left|\sin\left(x - \frac{\pi}{4}\right)\right| + c$$

D

$$x - \log\left|\cos\left(x - \frac{\pi}{4}\right)\right| + c$$