Chapter 5 - Section 5.5 - Indefinite Integrals and the Substitution Method - Exercises - Page 330: 55

$$\int \frac{dx}{x\ln x}=\ln|\ln x|+C$$

$$A=\int \frac{dx}{x\ln x}=\int\Big(\frac{1}{\ln x}\times\frac{dx}{x}\Big)$$ We set $u=\ln x$ Then $$du=\frac{dx}{x}$$ Therefore, $$A=\int\frac{1}{u}du=\ln|u|+C$$ $$A=\ln|\ln x|+C$$