First slide

Evaluation of definite integrals

Question

The value of the integral $\int_{1 / n}^{(an - 1) / n} \frac{\sqrt{x}}{\sqrt{a - x} + \sqrt{x}} dx$ is

Moderate

A

$\frac{a}{2}$
B

$\frac{na + 2}{2 n}$
C

$\frac{na - 2}{2 n}$
D

None of these

Solution

$\begin{array}{l} I = \int_{1 / n}^{(an - 1) / n} \frac{\sqrt{x}}{\sqrt{a - x} + \sqrt{x}} dx \\ = \int_{1 / n}^{a - (1 / n)} \frac{\sqrt{x}}{\sqrt{a - x} + \sqrt{x}} dx - - - - (i) \\ = \int_{1 / n}^{a - (1 / n)} \frac{\sqrt{\frac{1}{n} + a - \frac{1}{n} - x} dx}{\sqrt{a - (\frac{1}{n} + a - \frac{1}{n} - x)} + \sqrt{(\frac{1}{n} + a - \frac{1}{n} - x)}} \\ \Rightarrow I = \int_{\frac{1}{n}}^{a - \frac{1}{n}} \frac{\sqrt{a - x}}{\sqrt{x} + \sqrt{a - x}} dx - - - - (ii) \end{array}$
On adding Eqs. (1) and (ii), we get
$\begin{array}{l} 2 I = \int_{1 / n}^{a - (1 / n)} 1 dx = [x]_{\frac{1}{n}}^{a - \frac{1}{n}} \\ \Rightarrow 2 I = a - \frac{1}{n} - \frac{1}{n} = \frac{na - 2}{n} \Rightarrow I = \frac{na - 2}{2 n} \end{array}$

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