Primitive of Reciprocal

Theorem

 * $\displaystyle \int \frac {\mathrm dx} x = \ln |x| + C$

for $x \ne 0$.

Corollary

 * $\displaystyle \frac {\mathrm d}{\mathrm dx}\ln |x| = \frac 1 x$

for $x \ne 0$.

Proof
Suppose $x > 0$.

Then $\ln |x| = \ln x$ and the result follows directly from Derivative of Natural Logarithm Function and the definition of indefinite integral.

Suppose $x < 0$.

Then:

and the result again follows from the definition of the indefinite integral.