$\int \sqrt{x + \sqrt{(x^{2} + 1)}} d x equals ( where C is integration Constant) to$

Moderate

A

$\frac{1}{3} {\{x + \sqrt{x^{2} + 1}\}}^{\frac{3}{2}} - {\{x + \sqrt{x^{2} + 1}\}}^{\frac{- 1}{2}} + C$
B

$\frac{1}{2} {\{x + \sqrt{x^{2} + 1}\}}^{\frac{1}{2}} - {\{x + \sqrt{x^{2} + 1}\}}^{\frac{- 1}{2}} + C$
C

$\frac{1}{3} {\{x - \sqrt{x^{2} + 1}\}}^{\frac{3}{2}} - {\{x + \sqrt{x^{2} + 1}\}}^{\frac{1}{2}} + C$
D

$\frac{1}{3} {\{x + \sqrt{x^{2} + 1}\}}^{\frac{3}{2}} + {\{x + \sqrt{x^{2} + 1}\}}^{\frac{- 1}{2}} + C$

Solution

$\begin{array}{l} Let I = \int \sqrt{x + \sqrt{x^{2} + 1}} d x \\ Let x + \sqrt{x^{2} + 1} = t \dots . (1) \\ Rationalize \Rightarrow \sqrt{x^{2} + 1} - x = \frac{1}{t} \dots . (2) \\ Subtract (1) & (2) \\ \Rightarrow 2 x = t - \frac{1}{t} \\ \Rightarrow x = \frac{1}{2} (t - \frac{1}{t}) \end{array}$
Differentiating with respect to $x$ on both sides
$\begin{array}{l} d x = \frac{1}{2} (1 + \frac{1}{t^{2}}) d t \\ ∴ I = \frac{1}{2} \int (t^{\frac{1}{2}} + t^{- \frac{3}{2}}) d t \\ = \frac{1}{3} t^{\frac{3}{2}} - t^{- \frac{1}{2}} + C \end{array}$
, $w h e r e t = x + \sqrt{x^{2} + 1}$

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