Primitive of Reciprocal

Theorem

 * $\ds \int \frac {\d x} x = \ln \size x + C$

for $x \ne 0$.

Proof
Suppose $x > 0$.

Then:
 * $\ln \size x = \ln x$

The result follows from Derivative of Natural Logarithm Function and the definition of primitive.

Suppose $x < 0$.

Then:

and the result again follows from the definition of the primitive.

Also presented as
Some sources gloss over the case where $x < 0$ and merely present this result as:


 * $\ds \int \frac {\d x} x = \ln x + C$

As far as is concerned, that is a mistake.