Practicing Success
The value of $\int\limits_{1 / n}^{(a n-1) / n} \frac{\sqrt{x}}{\sqrt{a-x}+\sqrt{x}} d x$, is |
$\frac{a}{2}$ $\frac{1}{2 n}(n a+2)$ $\frac{n a-2}{2 n}$ none of these |
$\frac{n a-2}{2 n}$ |
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 $2 I=\int\limits_{1 / n}^{(a n-1) / n} 1 . d x=\frac{a n-2}{n} \Rightarrow I=\frac{a n-2}{2 n}$ |