Primitive of Reciprocal

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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.

$\blacksquare$


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