Primitive of Reciprocal/Corollary
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Corollary to Primitive of Reciprocal
- $\ds \int \frac {\d x} x = \ln x + C$
for $x > 0$.
Proof
From Primitive of Reciprocal:
- $\ds \int \frac {\d x} x = \ln \size x + C$
for $x \ne 0$.
By definition of absolute value:
- $\forall x \in \R_{>0}: \size x = x$
Hence the result.
$\blacksquare$