DefinitionDefinition: Literature-sourcedProof: Not applicableSession-authored (Fable 5 assisted)audited 2026-08-13

How statement and proof provenance work

The first chip identifies the source of the statement or construction; the second identifies the source of its local proof or verification.

Literature-sourced: the exact statement appears in a cited source; only wording and notation differ.
AI-adapted: a semantically identical restatement of literature-sourced material, modulo indexing, notation, and boundary cases adopted by the library.
AI-generated: a genuinely novel statement formulated by AI, with no source for the claim itself.

These labels describe origin, not correctness: citations and verification chips remain separate evidence.

The integral logarithm $L (x) : = \int_{1}^{x} \frac{d t}{t}$ for $x > 0$

Definition

For $x > 0$ , define the integral logarithm

$L (x) : = \int_{1}^{x} \frac{d t}{t},$

using the oriented integral when $x < 1$ (The integral with oriented limits: $\int_{a}^{a} f : = 0$ and $\int_{b}^{a} f : = - \int_{a}^{b} f$ ).

This is well defined. The function $t \mapsto 1 / t$ is continuous wherever $t \neq 0$ by the quotient clause of Sums, scalar multiples, products, absolute values, maxima, minima and quotients with nonvanishing denominator of continuous functions are continuous, as are constants, the identity and every polynomial function. For $x \neq 1$ it is therefore continuous on the nondegenerate compact interval with endpoints $1$ and $x$ , hence Riemann integrable there by A continuous function on $[a, b]$ is Riemann integrable, by Heine-Cantor and Riemann's criterion, whose hypothesis is $a < b$ . At $x = 1$ the interval is degenerate and that theorem does not apply; there The integral with oriented limits: $\int_{a}^{a} f : = 0$ and $\int_{b}^{a} f : = - \int_{a}^{b} f$ stipulates $\int_{1}^{1} f = 0$ , so $L (1) = 0$ directly.

Depends on

Used by

The integral logarithm satisfies L'(x)=1/x for x>0 and L(1)=0 Theorem

Dependency tree · next 3 levels

Direct dependencies and their dependencies through the next three levels: 81 results over 16 levels. An arrow runs from a result to what uses it, and this result sits at the bottom with a heavier outline. Click the chart to enlarge it.

Sources

OpenStax, Calculus Volume 1, Section 6.7 (standard reference, not scraped)

The integral logarithm L(x):=∫1xdttL(x):=∫1x​tdt​ for x>0x>0