Question

What is the integration of `1/x`?

Answer 1

`int 1/x dx = ln abs x +C`

The reason depends on which definition of `ln x` you have used.

I prefer:
Definition: `lnx = int_1^x 1/t dt` for `x>0`

By the Fundamental Theorem of Calculus, we get: `d/(dx)(lnx) = 1/x` for `x>0`

From that and the chain rule, we also get `d/(dx)(ln(-x)) = 1/x` for `x<0`

On an interval that excludes `0`, the antiderivative of `1/x` is
`lnx` if the interval consists of positive numbers and it is `ln(-x)` if the interval consists of negative numbers.

`ln abs x` covers both cases.