# Polynomial antiderivatives and the choice of constant

This English edition is an AI candidate and does not yet have an OCME Atlas (GPT-6) Twin closure receipt.

An antiderivative differentiates to f on its declared interval; this pulse selects a zero-constant rational-coefficient representative for full-real polynomials.

mko-antiderivative · definition · v0.10.0



## Mathematical statement

F is an antiderivative of f on an interval when F is differentiable there and F′=f. For the full-real polynomial f(x)=Σa_k*x^k, choose F0(x)=Σa_k*x^(k+1)/(k+1) with constant term zero; F0+C is also an antiderivative. Within the polynomial model, identical derivative coefficients force identical positive-degree coefficients, leaving only the constant term free.





## Representative formula (exact TeX)

A(f)





## Assumptions


- Polynomial f uses one through four ascending integer coefficients; a candidate F uses one through five ascending rational pairs [n,d] with d>0. Do not infer F′=f from finitely many evaluation samples.
- Monomials c*x^n separately allow n from zero through five, with a nonzero primitive list up to length seven. Polynomial x^0 is one even at x=0. At c=0, normalize the zero primitive to [0/1] rather than retaining meaningless higher zero terms.
- Representative answers list reduced positive-denominator rational coefficients in ascending order with constant zero. When evaluating F0(x)+C, retain actual C and x even though differentiation removes C.





## Symbol scope


- A — scoped zero-constant polynomial-primitive representative
- f — specified full-real polynomial





## Intuition, definition, and misconceptions

One antiderivative and all antiderivatives are different outputs. F0 is a reproducible representative, but the actual family F0+C is retained.

Check candidate derivatives coefficient by coefficient against f. The constant term does not affect that identity, but affects function evaluation. Antiderivatives of the zero polynomial are constants, not arbitrary fictitious higher-degree representatives.

The difference-by-a-constant check here is polynomial algebra. General differentiable functions on an interval need a separate mean-value-theorem obligation, and the conclusion is not transferred to a disconnected domain. An antiderivative is neither an inverse function nor a reciprocal derivative value.

Common errors include dropping the constant family, replacing polynomial identity with a few matching values, or treating an antiderivative as an inverse function.





## Adopted derivation guide

[Antiderivatives: the representative polynomial antiderivative, the constant family and checking candidates](https://ocme.evemisslab.com/en/derivations/derivation-mko-antiderivative-01/)The original MKO is retained. The guide has its own append-only revision and verification boundaries, not a new Lean proof.





## Prerequisites


- [Real derivatives at a point and finite difference quotients](https://ocme.evemisslab.com/en/mko/mko-derivative/) — Use the antiderivative criterion F′=f.
- [Differentiating polynomial sums, products, and compositions](https://ocme.evemisslab.com/en/mko/mko-derivative-rules/) — Use polynomial derivatives and coefficient identity.
- [Rational number](https://ocme.evemisslab.com/en/mko/mko-rational-number/) — Represent division by k+1 exactly.





## Verification boundary

Finite coefficients and conditional evaluation can be recomputed; general antiderivative existence and interval uniqueness up to a constant have no new formal Evidence.

not_formalized

This does not mean the content is false; it means no replayable computational or formal evidence is registered yet.





## Next obligation

Bridge rational-coefficient polynomial derivative identity and a selected constant, retaining the full-real interval and general-existence boundary.





## Same-source structured data

This page and the interactive view use the same object and existing language edition; this adds no proof or human review.

[Canonical MKO JSON](https://ocme.evemisslab.com/data/mko/antiderivative.json) · [Plain Markdown](https://ocme.evemisslab.com/en/mko/mko-antiderivative/index.md)