Integral can't find how to do it: $\int\frac{2\ln(x)}{x}dx$

Question

I have to find this integral $$\int\frac{2\ln(x)}{x}dx$$ This is how I began: $$\int\frac{2\ln(x)}{x}dx=2\int\frac{\ln(x)}{x}dx$$ Then I tried substitution $e^u=x$ to get $u=\ln(x)\longrightarrow du=\frac1{dx}\rightarrow dx=\frac1{du}$

$$2\int\frac{\ln(x)}{x}dx=2\int\frac{\ln(e^u)}{e^u}\frac1{dx}$$ But then it gets complicated, and I'm not sure if the steps I've done are right. Need some help guys.

Answer 1

$$\ln x=u\Rightarrow \frac{1}{x}dx=du\Rightarrow\int\frac{2\ln(x)}{x}dx=2\int udu=u^2+c=\ln^2x+c$$

Answer 2

$$\ln x=y\implies \frac{dx}x=dy\implies\int\frac{\ln x}x\ dx=\int y\ dy$$

Answer 3

It should be $du = \frac1x dx $

Answer 4

Hint :

Rewrite the integral as $$ 2\int\ln x\ d(\ln x). $$