Explanation

(b)
Given, $\frac{dy}{dx} + \frac{y}{(1 - x)\sqrt{x}} = 1 - \sqrt{x}$
$\therefore$ $IF = e^{\int_{}^{}{\frac{1}{(1 - x)\sqrt{x}}\ dx}}$
Put $\sqrt{x} = t$
$\Rightarrow$ $\frac{1}{2\sqrt{x}}\ dx = dt$
$\therefore$ $IF = e^{\int_{}^{}{\frac{2}{1 - t^{2}}\ dt}}$
$= e^{\frac{2}{2}\log\left( \frac{1 + t}{1 - t} \right)} = \frac{1 + t}{1 - t} = \frac{1 + \sqrt{x}}{1 - \sqrt{x}}$