First slide

Introduction to integration

Question

$\int \frac{l o g (x + \sqrt{1 + x^{2}})}{\sqrt{1 + x^{2}}} d x$

Moderate

A

$\frac{1}{2} {[\log (x + \sqrt{1 + x^{2}})]}^{2} + c$
B

${[\log (x + \sqrt{1 + x^{2}})]}^{2} + c$
C

$[\log (x + \sqrt{1 + x^{2}})] + c$
D

$N o n e o f t h e s e$

Solution

$\begin{array}{rcl} \begin{array}{l} L e t l o g (x + \sqrt{1 + x^{2}}) = t \\ D i f f e r e n t i a t i n g, w e g e t \frac{1}{x + \sqrt{1 + x^{2}}} (1 + \frac{2 x}{2 \sqrt{1 + x^{2}}}) d x = d t \\ \Rightarrow \frac{1}{\sqrt{1 + x^{2}}} d x = d t \\ ∴ \int \frac{l o g (x + \sqrt{1 + x^{2}})}{\sqrt{1 + x^{2}}} d x = \int t d t \\ = \frac{t^{2}}{2} + c \\ = \frac{1}{2} {[\log (x + \sqrt{1 + x^{2}})]}^{2} + c \end{array} \end{array}$

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