Explanation

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