Question

$\int \frac{1n|x|}{|x|\sqrt{1+1n|x|}}dx$ equals:

• A. $\frac{2}{3}\sqrt{1+1n|x|}(ln|x|-2)+c$
• B. $\frac{2}{3}\sqrt{1+1n|x|}(ln|x|+2)+c$
• C. $\frac{1}{3}\sqrt{1+1n|x|}(ln|x|-2)+c$
• D. $2\sqrt{1+1n|x|}(3ln|x|-2)+c$

Answer 1

The correct option is A $\frac{2}{3}\sqrt{1+1n|x|}(ln|x|-2)+c$

$I=\int \frac{ln|x|}{|x|\sqrt{1+ln|x|}}dx$

Put; $1+ln|x|=t^2\Rightarrow ln|x|=t^2-1$

$\frac{1}{|x|}dx=2tdt$

$I=\int \frac{(t^2-1)2tdt}{t}=2\int t^2-1dt$

$=2[\frac{t^3}{3}-t]+c$

$=\frac{2}{3}(1+ln|x|)\sqrt{1+ln|x|}-2\sqrt{1+ln|x|}+c$

$=\frac{2}{3}\sqrt{1+ln|x|}[1+ln|x|-\frac{3}{2}\times 2]+c$

$=\frac{2}{3}\sqrt{1+ln|x|}[ln|x|-2]+c$