Primitive of Reciprocal/Proof

Theorem

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

for $x \ne 0$.

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:

$\ds \dfrac \d {\d x} \ln \size x$

$=$

$\ds \dfrac \d {\d x} \map \ln {-x}$

Definition of Absolute Value

$\ds $

$=$

$\ds \frac 1 {-x} \cdot -1$

$\ds $

$=$

$\ds \frac 1 x$

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

$\blacksquare$

1944: R.P. Gillespie: Integration (2nd ed.) ... (previous) ... (next): Chapter $\text {II}$: Integration of Elementary Functions: $\S 7$. Standard Integrals: $2$.
1953: L. Harwood Clarke: A Note Book in Pure Mathematics ... (previous) ... (next): $\text {II}$. Calculus: Integration
1967: Michael Spivak: Calculus ... (previous) ... (next): Part $\text {III}$: Derivatives and Integrals: Chapter $18$: Integration in Elementary Terms
1972: Frank Ayres, Jr. and J.C. Ault: Theory and Problems of Differential and Integral Calculus (SI ed.) ... (previous) ... (next): Chapter $25$: Fundamental Integration Formulas: $5$.
1974: Murray R. Spiegel: Theory and Problems of Advanced Calculus (SI ed.) ... (previous) ... (next): Chapter $5$. Integrals: Integrals of Special Functions: $2$