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I am trying to gain a deeper understanding of the integral of 1/x.

I understand that ln(a) is the area under graph 1/x from x=1 to x=a, where a>0, this is a definite integral.

What I am trying to wrap my head around is the integral of 1/x being ln|x|, with the absolute value of x causing me the most confusion. How does this relate to the above definition? What does this indefinite integral mean?

I have seen it explained that for y = ln(-x) where x<0, by chain rule dy/dx = -1/(-x) = 1/x, thus the integral of 1/x is ln|x|, but it is here that I am losing all intuition of the concept. I understand the process of using the chain rule and how this solution is arrived at, but I'm left wondering what the indefinite integral of 1/x really is as a mentally tangible concept.