Try to compute $$\int\frac{dx}{x\ln x}$$ I compute it this way: first we have $x>0$. \begin{align*} \int\frac{dx}{x\ln x} &=\int\frac{d(\ln x)}{\ln x}\\ &=\ln|\ln x|+C \end{align*} But the answer to the problem is $\ln\ln x+C$. Which one is right? Thanks!
Source Григорий Михайлович Фихтенгольц
Both and none. Your solution is a primitive both on $(0,1)$ and on $(1,+\infty)$. The answer of the book is a primitive only on $(1,+\infty)$. Therefore, I'd say that your answer is better and more complete.