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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

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up vote 5 down vote accepted

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.

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Although the constant of integration can be different on each of $(0,1)$ and $(1, \infty)$. – wckronholm Jul 2 '12 at 15:48
Yes, of course. Anyway, both $(0,1)$ and $(1,+\infty)$ are connected sets, so one primitive is enough. – Siminore Jul 2 '12 at 17:14

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