Primitive of Reciprocal

Theorem

 * $\displaystyle \int \frac {\mathrm dx} x = \ln \left\vert{x}\right\vert + C$

for $x \ne 0$.

Proof
Suppose $x > 0$.

Then:
 * $\ln \left\vert{x}\right\vert = \ln x$

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.