Explanation

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