Question

$\int \frac{ln\left(x\right)}{x\sqrt{1+ln\left(x\right)}}dx$ equlas

• A. $\frac{2}{3}\sqrt{1+ln\left|x\right|}(ln\left|x\right|-2)+c$
• B. $\frac{4}{3}\sqrt{1+ln\left|x\right|}(ln\left|x\right|-2)-c$
• C. $\frac{1}{3}\sqrt{1+ln\left|x\right|}(ln\left|x\right|-2)+c$
• D. $2\sqrt{1+ln\left|x\right|}(3ln\left|x\right|-2)+c$

Answer 1

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

Let $I=\int \frac{\ln \left(x\right)}{x\sqrt{1+\ln x}}dx$
Substitute $t=\ln x\Rightarrow dt=\frac{1}{x}dx$

$\therefore I=\int \frac{t}{\sqrt{t+1}}dt$

Substitute $t=u+1\Rightarrow dt=du$

$\therefore I=\int \frac{u-1}{\sqrt{u}}du=\int (\sqrt{u}-\frac{1}{\sqrt{u}})du$
$=\frac{{2u}^{\frac{2}{3}}}{3}-2\sqrt{u}+c$
$=\frac{2}{3}{\left(\ln x-2\right)}^{\frac{3}{2}}-2\sqrt{\ln x+1}$
$=\frac{2}{3}(\ln |x|-2)\sqrt{\ln \left|x\right|+1}$