Explanation

(d)
Let $I = \int_{0}^{1}\frac{dx}{x + \sqrt{1 - x^{2}}}$
Put $x = \cos{\theta \Rightarrow dx = - \sin{\theta d\theta}}$
$\therefore I = \int_{\pi/2}^{0}\frac{- \sin{\theta d\theta}}{\cos{\theta +}\sqrt{1 - \cos^{2}\theta}}$
$= \int_{0}^{\pi/2}{\frac{\sin\theta}{\sin{\theta + \cos\theta}}d\theta}$……(i)
$\Rightarrow I = \int_{0}^{\pi/2}{\frac{\sin\left( \frac{\pi}{2} - \theta \right)}{\sin{\left( \frac{\pi}{2} - \theta \right) + \cos\left( \frac{\pi}{2} - \theta \right)}}d\theta}$
$= \int_{0}^{\pi/2}{\frac{\cos\theta}{\sin{\theta + \cos\theta}}d\theta}$…(ii)
On adding Eqs. (i) and (ii), we get
$2I = \int_{0}^{\pi/2}{d\theta = \frac{\pi}{2}}$
$\Rightarrow I = \frac{\pi}{4}$