Primitive of Reciprocal/Corollary

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Corollary to Primitive of Reciprocal

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

for $x > 0$.


Proof

From Primitive of Reciprocal:

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

for $x \ne 0$.

By definition of absolute value:

$\forall x \in \R_{>0}: \size x = x$

Hence the result.

$\blacksquare$