Explanation

(a)
Let $I = \int_{}^{}{\sqrt{2}\ \sqrt{1 + \sin{x\ }}dx = \sqrt{2}\int_{}^{}{\left( \sin{\frac{x}{2} + \cos\frac{x}{2}} \right)dx}}$
$= 2\int_{}^{}{\sin{\left( \frac{\pi}{4} + \frac{x}{2}\ \right)dx = - 4\cos{\left( \frac{x}{2} + \frac{\pi}{4}\ \right) + c}}}$
But $I = - 4\cos{(ax + b) + c}$ (given)
On comparing, we get
$a = \frac{1}{2},\ b = \frac{\pi}{4}$