Explanation

(c)
Let $I = \left\lbrack \sin{x + \cos x} \right\rbrack^{1 + \sin{2x}}$
$= \left\lbrack \sqrt{2}\sin\left( \frac{\pi}{4} + x \right) \right\rbrack^{1 + \sin{2x}}$
At $x = \frac{\pi}{4},$
$I = \left\lbrack \sqrt{2}\sin\left( \frac{\pi}{4} + \frac{\pi}{4} \right) \right\rbrack^{1 + \sin\frac{2\pi}{4}}$
$= \left( \sqrt{2} \right)^{2} = 2$