Explanation

(c)
$I = \int_{1/n}^{(an - 1)/n}{\frac{\sqrt{x}}{\sqrt{a - x} + \sqrt{x}}dx}$
$= \int_{1/n}^{a - (1/n)}{\frac{\sqrt{x}}{\sqrt{a - x} + \sqrt{x}}dx}$ …(i)
$= \int_{1/n}^{a - (1/n)}\frac{\sqrt{\frac{1}{n} + a - \frac{1}{n} - x}dx}{\sqrt{a - \left( \frac{1}{n} + a - \frac{1}{n} - x \right)} + \sqrt{\left( \frac{1}{n} + a - \frac{1}{n} - x \right)}}$
$\Rightarrow I = \int_{\frac{1}{n}}^{a - \frac{1}{n}}\frac{\sqrt{a - x}}{\sqrt{x} + \sqrt{a - x}}dx$ …(ii)
On adding Eqs.(i) and (ii), we get
$2I = \int_{1/n}^{a - (1/n)}{1\ dx = \lbrack x\rbrack_{\frac{1}{n}}^{a - \frac{1}{n}}}$
$\Rightarrow 2I = a - \frac{1}{n} - \frac{1}{n} = \frac{na - 2}{n}$
$\Rightarrow I = \frac{na - 2}{2n}$