First slide

Methods of integration

Question

The value of $\int \frac{\log_{e} (x + \sqrt{x^{2} + 1})}{\sqrt{x^{2} + 1}} d x$ , is

Moderate

A

$2 \log_{e} (x + \sqrt{x^{2} + 1}) + C$
B

${\{\log_{e} (x + \sqrt{x^{2} + 1})\}}^{2} + C$
C

$\log (x + \sqrt{x^{2} + 1}) + C$
D

none of these

Solution

We have.
$\begin{array}{l} I = \int \frac{\log_{c} (x + \sqrt{x^{2} + 1})}{\sqrt{x^{2} + 1}} d x ∣ \\ \Rightarrow l = \int \log_{c} (x + \sqrt{x^{2} + 1}) d \{\log_{e} (x + \sqrt{x^{2} + 1})\} \\ \Rightarrow I = \frac{1}{2} {\{\log_{p} (x + \sqrt{x^{2} + 1})\}}^{2} + C \end{array}$

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