Explanation

(c)
Let $I = 100\int_{0}^{\pi/2}{\frac{(\sin{x + \cos{x) + \sin x}}}{\sin{x + \cos x}}dx}$
$= 100\left\lbrack \int_{0}^{\pi/2}{1dx + \int_{0}^{\pi/2}{\frac{\sin x}{\sin{x + \cos x}}dx}} \right\rbrack$
Let $I_{1} = \int_{0}^{\frac{\pi}{2}}\frac{\sin x}{\sin{x + \cos x}}$ …(i)
$\Rightarrow \ I_{1} = \int_{0}^{\pi/2}{\frac{\cos x}{\sin{x + \cos x}}dx}$ …(ii)
$\left\lbrack put\ x = \frac{\pi}{2} - \ x \right\rbrack$
On adding Eqs. (i) and (ii), we get
$2I_{1} = \int_{0}^{\pi/2}{1\ dx = \frac{\pi}{2}}$
$\Rightarrow \ I_{1} = \frac{\pi}{4}$
$\therefore\ I = 100\left\lbrack \frac{\pi}{2} + \frac{\pi}{4} \right\rbrack = 100 \times \frac{3\pi}{4} = 75\pi$