Primitive of Reciprocal/Proof
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Theorem
- $\ds \int \frac {\d x} x = \ln \size x + C$
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:
\(\ds \dfrac \d {\d x} \ln \size x\) | \(=\) | \(\ds \dfrac \d {\d x} \map \ln {-x}\) | Definition of Absolute Value | |||||||||||
\(\ds \) | \(=\) | \(\ds \frac 1 {-x} \cdot -1\) | Chain Rule for Derivatives and Derivative of Natural Logarithm Function, as $-x > 0$ | |||||||||||
\(\ds \) | \(=\) | \(\ds \frac 1 x\) |
and the result again follows from the definition of the primitive.
$\blacksquare$
Sources
- 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: $2$