Question 1305: Zero-Constant Antiderivative of a Monomial

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Original question and givens
Definition and scope
Step-by-step derivation
Reverse checks
Result
This question review and source binding

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R0001-01305 · q-20261007-advance-20261008-r06-antiderivative-0005

Original question and givens

On all R, f(x)=(5)x^4, 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)=(5)x^4, 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

Step-by-step derivation

S001 · implies

Before

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 dependencies

Rule: polynomial_representation · Uses: G001

S002 · implies

Before

f coefficients ascending=[0, 0, 0, 0, 5]

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 dependencies

Rule: canonical_primitive_domain · Uses: G001, S001

S003 · calculation

Before

primitive coefficient at degree1 := div(0, 1)

After

primitive coefficient at degree1 = 0

Conditions: none

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

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S002

S004 · calculation

Before

differentiate back coefficient0 := mul(0, 1)

After

differentiate back coefficient0 = 0

Conditions: none

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

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S003

S005 · calculation

Before

primitive coefficient at degree2 := div(0, 2)

After

primitive coefficient at degree2 = 0

Conditions: none

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

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S004

S006 · calculation

Before

differentiate back coefficient1 := mul(0, 2)

After

differentiate back coefficient1 = 0

Conditions: none

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

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S005

S007 · calculation

Before

primitive coefficient at degree3 := div(0, 3)

After

primitive coefficient at degree3 = 0

Conditions: none

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

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S006

S008 · calculation

Before

differentiate back coefficient2 := mul(0, 3)

After

differentiate back coefficient2 = 0

Conditions: none

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

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S007

S009 · calculation

Before

primitive coefficient at degree4 := div(0, 4)

After

primitive coefficient at degree4 = 0

Conditions: none

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

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S008

S010 · calculation

Before

differentiate back coefficient3 := mul(0, 4)

After

differentiate back coefficient3 = 0

Conditions: none

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

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S009

S011 · calculation

Before

primitive coefficient at degree5 := div(5, 5)

After

primitive coefficient at degree5 = 1

Conditions: none

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

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S010

S012 · calculation

Before

differentiate back coefficient4 := mul(1, 5)

After

differentiate back coefficient4 = 5

Conditions: none

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

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S011

S013 · implies

Before

[0, 0, 0, 0, 0, 1]

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 dependencies

Rule: global_polynomial_identity · Uses: G001, S012

S014 · implies

Before

[0, 0, 0, 0, 0, 1]

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 dependencies

Rule: coefficient_output_protocol · Uses: G001, S013

Reverse checks

S014 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
S013 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
S012 · premise_and_result_audit — Recompute differentiate back coefficient4 with exact rational arithmetic, not rounded decimals.
S011 · premise_and_result_audit — Recompute primitive coefficient at degree5 with exact rational arithmetic, not rounded decimals.
S010 · premise_and_result_audit — Recompute differentiate back coefficient3 with exact rational arithmetic, not rounded decimals.
S009 · premise_and_result_audit — Recompute primitive coefficient at degree4 with exact rational arithmetic, not rounded decimals.
S008 · premise_and_result_audit — Recompute differentiate back coefficient2 with exact rational arithmetic, not rounded decimals.
S007 · premise_and_result_audit — Recompute primitive coefficient at degree3 with exact rational arithmetic, not rounded decimals.
S006 · premise_and_result_audit — Recompute differentiate back coefficient1 with exact rational arithmetic, not rounded decimals.
S005 · premise_and_result_audit — Recompute primitive coefficient at degree2 with exact rational arithmetic, not rounded decimals.
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, 0/1, 0/1, 0/1, 0/1, 1/1]

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: b46553a6e82f804f0abffa30a3e914556fc46bd7d51c88148be6904a4634b4a9
candidate record SHA-256: f8be29afe8870aa2fc81bc2674e3c04c6fac10f0935713e87d37e752d698a15a
semantic review receipt SHA-256: c3b44ce8ed60ada6902fb206627537d5f90ba2a867eaba43a36f903280c496cc
question-first witness SHA-256: 3e91602adc7f81a2671a2a3ffaf6a447028b56abc5853e8271673d76f30cd7ad
Lean: NOT_RUN · human review: NOT_PERFORMED

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