CorollaryStatement: Literature-sourcedProof: AI-adaptedSession-authored (Fable 5 assisted)precheck passjudge pass (deepseek-v4-pro + gpt-5.6-terra)audited 2026-08-21

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Henstock–Kurzweil substitution for a derivative composed with a differentiable map

Statement

Let $a < b$ , let $ϕ : [a, b] \to R$ be differentiable, let $J$ be a nondegenerate interval containing $ϕ ([a, b])$ , and let $F : J \to R$ be differentiable with $F^{'} = f$ . Then $(f \circ ϕ) ϕ^{'}$ is Henstock–Kurzweil integrable and

$\int_{a}^{b} f (ϕ (t)) ϕ^{'} (t) d t = F (ϕ (b)) - F (ϕ (a)) .$

No monotonicity of $ϕ$ is required.

Facts & Assumptions

Given: The functions and the containing interval in the Statement.

[L1]

The chain rule gives $(F \circ ϕ)^{'} = (F^{'} \circ ϕ) ϕ^{'}$ (The chain rule, in one line from Carathéodory: if $g$ is differentiable at $c$ and $f$ is differentiable at $g (c)$ , then $f \circ g$ is differentiable at $c$ with $(f \circ g)^{'} (c) = f^{'} (g (c)) g^{'} (c)$ ).

[L2]

Every derivative is Henstock–Kurzweil integrable and evaluates by endpoint difference (Every derivative is Henstock–Kurzweil integrable and satisfies Newton–Leibniz).

Proof

technique · direct

1.1

Applying [L1] throughout $[a, b]$ identifies the derivative of $F \circ ϕ$ as $(f \circ ϕ) ϕ^{'}$ , including a constant $ϕ$ .

givenL1

2.1

Applying [L2] to the composite gives its HK integrability and the displayed endpoint formula, which also covers reversed endpoint values of $ϕ$ .

step 1.1L2∎

Depends on

Used by

Henstock–Kurzweil integrability does not imply integrability of the absolute value Counterexample

Dependency tree · two levels

14 results within two dependency steps of this one, each drawn at its shortest distance from it. An arrow runs from a result to what uses it, so the chart reads left to right and ends at this result, which carries a heavier outline. Every node is a link to that result. Click elsewhere on the chart to enlarge it.

Sources

Alessandro Fonda, The Kurzweil-Henstock Integral for Undergraduates, Ch. 1 (standard reference, not scraped)
Andrew Bruckner, Judith Bruckner and Brian Thomson, Real Analysis, Section 1.21 (standard reference, not scraped)