# Primitive of Reciprocal

## Theorem

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

for $x \ne 0$.

### Corollary 1

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

for $x > 0$.

### Corollary 2

$\displaystyle \frac {\d} {\d x} \ln \size x = \frac 1 x$

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:

 $\displaystyle \dfrac \d {\d x} \ln \size x$ $=$ $\displaystyle \dfrac \d {\d x} \map \ln {-x}$ Definition of Absolute Value $\displaystyle$ $=$ $\displaystyle \frac 1 {-x} \cdot -1$ Chain Rule for Derivatives and Derivative of Natural Logarithm Function, as $-x > 0$ $\displaystyle$ $=$ $\displaystyle \frac 1 x$

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

$\blacksquare$