# Question 1319: Zero-Constant Antiderivative of a Monomial

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This teaching derivation has second-context AI mathematical and bilingual semantic review, limited to the model and assumptions stated in this question. It has no human review or new Lean proof and does not promote the original question or MKO to Canonical truth.

R0001-01319 · q-20261007-advance-20261008-r06-antiderivative-0019



## Original question and givens

On all R, f(x)=(19)x^0, with 0≤n≤5 and x^0=1 even at zero. Give ascending reduced rational coefficients of its zero-constant primitive.


- G001 — On all R, f(x)=(19)x^0, with 0≤n≤5 and x^0=1 even at zero. Give ascending reduced rational coefficients of its zero-constant primitive.





## Definition and scope

Use the original question domain, assumptions and cited MKO definitions. This item has second-context mathematical and bilingual review, with the model, answer and every arithmetic step independently recomputed from the original parameters; the source question remains a candidate.


- [mko-antiderivative](https://ocme.evemisslab.com/en/mko/mko-antiderivative/)





## Step-by-step derivation

S001 · impliesBefore

domain=R; coefficient lists are in ascending powers

After

constant term first; rational pairs=[numerator,denominator]

Conditions: none

Fix coefficient order;do not read ascending coefficients as descending or swap numerator and denominator.

Step identifiers and dependenciesRule: polynomial_representation · Uses: G001





S002 · impliesBefore

f coefficients ascending=[19]

After

seek F₀ with constant coefficient0 and F₀′=f

Conditions: none

Antiderivatives are not generally unique; this task additionally fixes the constant coefficient at0.

Step identifiers and dependenciesRule: canonical_primitive_domain · Uses: G001, S001





S003 · calculationBefore

primitive coefficient at degree1 := div(19, 1)

After

primitive coefficient at degree1 = 19

Conditions: none

Since (x^(k+1))′=(k+1)x^k,divide the coefficient by the positive nonzero integer k+1.

Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S002





S004 · calculationBefore

differentiate back coefficient0 := mul(19, 1)

After

differentiate back coefficient0 = 19

Conditions: none

Differentiation multiplies back by k+1 and must recover the original coefficient.

Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S003





S005 · impliesBefore

[0, 19]

After

all derivative coefficients agree with f

Conditions: none

Coefficient agreement proves the polynomial identity on all reals,not merely equality at a few points.

Step identifiers and dependenciesRule: global_polynomial_identity · Uses: G001, S004





S006 · impliesBefore

[0, 19]

After

reduced coefficients;integer coefficients retain /1

Conditions: none

Return ascending coefficients,trim redundant leading-in-degree zeros,and retain[0/1] for the zero polynomial.

Step identifiers and dependenciesRule: coefficient_output_protocol · Uses: G001, S005









## Reverse checks


- S006 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
- S005 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
- S004 · premise_and_result_audit — Recompute differentiate back coefficient0 with exact rational arithmetic, not rounded decimals.
- S003 · premise_and_result_audit — Recompute primitive coefficient at degree1 with exact rational arithmetic, not rounded decimals.
- S002 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
- S001 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.





## Result

[0/1, 19/1]

The full substitutions and premise checks give the original answer; source identity,format and status are unchanged.





## This question review and source binding

Review date: 2026-10-26

Original questions, answers, step expressions, numeric ASTs and historical records are retained. Source candidate labels remain historical and are not overwritten as formal proof.

Source hashes and review receipts
- input commit: 0f7be61e2c22317a4075561dc822c87d881a50db
- original question SHA-256: fe45be5de755711f4ac744d46bc4f981b0cad77732f29b21a5d591c177793508
- candidate record SHA-256: a2f64fed373b1b91537d301a99c180db7823ae19da1a62afb5b6bfe1914c8158
- semantic review receipt SHA-256: c3b44ce8ed60ada6902fb206627537d5f90ba2a867eaba43a36f903280c496cc
- question-first witness SHA-256: 3e91602adc7f81a2671a2a3ffaf6a447028b56abc5853e8271673d76f30cd7ad
- Lean: NOT_RUN · human review: NOT_PERFORMED

[Same-source full JSON](https://ocme.evemisslab.com/data/question-derivations/questions/q-20261007-advance-20261008-r06-antiderivative-0019.json)