First slide

Methods of integration

Question

The value of $f \int \frac{\sqrt{1 + x}}{x} d x$ , is

Moderate

A

$2 \sqrt{1 + x} + \log |\frac{\sqrt{1 + x} - 1}{\sqrt{1 + x} + 1}| + C$
B

$2 \sqrt{1 + x} + C$
C

$\log_{e} |\frac{\sqrt{1 + x} - 1}{\sqrt{1 + x} + 1}| + C$
D

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

Solution

Let
$I = \int \frac{\sqrt{1 + x}}{x} d x$
$\Rightarrow I = \int \frac{\sqrt{t^{2}}}{t^{2} - 1} - 2 t d t,$ where $1 + x = t^{2}$
$\begin{array}{l} \Rightarrow t = 2 \int \frac{t^{2} - 1 + 1}{t^{2} - 1} d t = 2 \int (1 + \frac{1}{t^{2} - 1}) d t \\ \Rightarrow I = 2 t + \log |\frac{t - 1}{t + 1}| + C = 2 \sqrt{1 + x} + \log |\frac{\sqrt{1 + x} - 1}{\sqrt{1 + x} + 1}| + C \end{array}$

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