# What is the integral of 1/x?

What is the integral of $$\frac{1}{x}\frac{1}{x}$$? Do you get $$\ln(x)\ln(x)$$ or $$\ln|x|\ln|x|$$?

In general, does integrating $$f'(x)/f(x)f'(x)/f(x)$$ give $$\ln(f(x))\ln(f(x))$$ or $$\ln|f(x)|\ln|f(x)|$$?

Also, what is the derivative of $$|f(x)||f(x)|$$? Is it $$f'(x)f'(x)$$ or $$|f'(x)||f'(x)|$$?

You have (Note that the “constant” $C$ might take different values for positive or negative $x$. It is really a locally constant function.)