Chapter 5 - Section 5.6 - Definite Integral Substitutions and the Area Between Curves - Exercises - Page 337: 30

$$\int^{4}_{2}\frac{dx}{x\ln x}=\ln\Big(\frac{\ln4}{\ln2}\Big)$$

$$A=\int^{4}_{2}\frac{dx}{x\ln x}$$ We set $u=\ln x$, which means $$du=\frac{1}{x}dx$$ For $x=4$, we have $$u=\ln4$$ For $x=2$, we have $$u=\ln2$$ Therefore, $$A=\int^{\ln4}_{\ln2}\frac{1}{u}du=\ln|u|\Big]^{\ln4}_{\ln2}$$ $$A=\ln(\ln4)-\ln(\ln2)$$ $$A=\ln\Big(\frac{\ln4}{\ln2}\Big)$$