Answer
$$\int\frac{dx}{x\log_{10}x}=\ln10(\ln|\ln x|)+C$$
Work Step by Step
$$A=\int\frac{dx}{x\log_{10}x}=\int\frac{dx}{\frac{x\ln x}{\ln10}}$$ $$A=\ln10\int\frac{dx}{x\ln x}$$
We set $u=\ln x$, which means $$du=\frac{dx}{x}$$
Therefore, $$A=\ln10\int\frac{1}{u}du$$ $$A=\ln10\times\ln|u|+C$$ $$A=\ln10(\ln|\ln x|)+C$$
