Primitive of Reciprocal/Corollary

Corollary to Primitive of Reciprocal

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

for $x > 0$.

Proof
From Primitive of Reciprocal:


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

for $x \ne 0$.

By definition of absolute value:
 * $\forall x \in \R_{>0}: \left\vert{x}\right\vert = x$

Hence the result.