For what value of a and b, the equation $\int(\sin 2 x-\cos 2 x) d x=\frac{1}{\sqrt{2}} \sin (2 x-a)+b$ holds good?

a = $-\frac{5 \pi}{4}$, b is any arbitrary constant

$\int(\sin 2 x-\cos 2 x) d x$

$=-\int \sqrt{2} \cos \left(2 x+\frac{\pi}{4}\right) d x$

$=\frac{-\sqrt{2}}{2} \sin \left(2 x+\frac{\pi}{4}\right)+c=\frac{1}{\sqrt{2}} \sin \left(2 x+\frac{5 \pi}{4}\right)+c$

a = $-\frac{5 \pi}{4}$, b is any arbitrary constant

Hence (1) is the correct answer.