Demystify integration of \(\displaystyle\int{\frac{{{1}}}{{{x}}}}{\left.{d}{x}\right.}\)

I've learned in my analysis class, that

\(\displaystyle\int{\frac{{{1}}}{{{x}}}}{\left.{d}{x}\right.}={\ln{{\left({x}\right)}}}\)

I can live with that, and it's what I use when solving equations like that. But how can I solve this, without knowing that beforehand.

Assuming the standard rule for integration is

\(\displaystyle\int{x}^{{a}}{\left.{d}{x}\right.}={\frac{{{1}}}{{{a}+{1}}}}\cdot{x}^{{{a}+{1}}}+{C}\)

If I use that and apply this to \(\displaystyle\int{\frac{{{1}}}{{{x}}}}{\left.{d}{x}\right.}\)

\(\displaystyle\int{\frac{{{1}}}{{{x}}}}{\left.{d}{x}\right.}=\int{x}^{{-{1}}}{\left.{d}{x}\right.}\)

\(\displaystyle={\frac{{{1}}}{{-{1}+{1}}}}\cdot{x}^{{-{1}+{1}}}\)

\(\displaystyle={\frac{{{x}^{{0}}}}{{{0}}}}\)

Obviously, this doesn't work, as I get a division by 0. I don't really see, how I can end up with ln(x). There seems to be something very fundamental that I'm missing.

I study computer sciences, so, we usually omit things like in-depth math theory like that. We just learned that \(\displaystyle\int{\frac{{{1}}}{{{x}}}}{\left.{d}{x}\right.}={\ln{{\left({x}\right)}}}\) and that's what we use.