Explanation

(d)
Let $I = \int_{0}^{\frac{\pi}{3}}\frac{\cos x + \sin x}{\sqrt{1 + 2\sin x\cos x}}dx$
$= \int_{0}^{\frac{\pi}{3}}\frac{\cos x + \sin x}{\sqrt{\cos^{2}x + \sin^{2}x + 2\sin{x\cos x}}}dx$
$= \int_{0}^{\frac{\pi}{3}}\frac{\cos x + \sin x}{\sqrt{\left( \cos x + \sin x \right)^{2}}}dx$
$= \int_{0}^{\frac{\pi}{3}}\frac{\cos x + \sin x}{\cos x + \sin x}dx$
$= \int_{0}^{\frac{\pi}{3}}1dx = \frac{\pi}{3}$