The value of ∫limits_1 / n^(a n-1) / n (√x)/(√(a-x)+√x) d x, is

Let

$I=\int\limits_{1 / n}^{(a n-1) / n} \frac{\sqrt{x}}{\sqrt{a-x}+\sqrt{x}} d x$       .....(i)

Using $\int\limits_a^b f(x) d x=\int\limits_a^b f(a+b-x) d x$, we have

$I=\int\limits_{1 / n}^{(a n-1) / n} \frac{\sqrt{a-x}}{\sqrt{x}+\sqrt{a-x}} d x$         ....(ii)

Adding (i) and (ii), we get

$2 I=\int\limits_{1 / n}^{(a n-1) / n} 1 . d x=\frac{a n-2}{n} \Rightarrow I=\frac{a n-2}{2 n}$