Question 1369: Coefficient Eligibility of a Candidate Antiderivative

Original question and givens

On all R, f has ascending coefficients [6, 19, -1, 3], starting with its constant term. The candidate F has ascending rational coefficient pairs [[5, 1], [6, 1], [19, 2], [-1, 3], [3, 4]], with positive denominators. Is F′=f as a coefficient identity?

• G001 — On all R, f has ascending coefficients [6, 19, -1, 3], starting with its constant term. The candidate F has ascending rational coefficient pairs [[5, 1], [6, 1], [19, 2], [-1, 3], [3, 4]], with positive denominators. Is F′=f as a coefficient identity?

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

domain=R; coefficient lists are in ascending powers

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

candidate F coefficients=[5, 6, 19/2, -1/3, 3/4]

target f coefficients=[6, 19, -1, 3]

candidate derivative coefficient0 := mul(1, 6)

candidate derivative coefficient0 = 6

candidate derivative coefficient1 := mul(2, 19/2)

candidate derivative coefficient1 = 19

candidate derivative coefficient2 := mul(3, -1/3)

candidate derivative coefficient2 = -1

candidate derivative coefficient3 := mul(4, 3/4)

candidate derivative coefficient3 = 3

differentiate all positive-degree terms

F′ coefficients=[6, 19, -1, 3]

F′ coefficients=[6, 19, -1, 3]; f coefficients=[6, 19, -1, 3]

F′=f identically: Yes

Reverse checks

• S008 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S007 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S006 · premise_and_result_audit — Recompute candidate derivative coefficient3 with exact rational arithmetic, not rounded decimals.
• S005 · premise_and_result_audit — Recompute candidate derivative coefficient2 with exact rational arithmetic, not rounded decimals.
• S004 · premise_and_result_audit — Recompute candidate derivative coefficient1 with exact rational arithmetic, not rounded decimals.
• S003 · premise_and_result_audit — Recompute candidate derivative coefficient0 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

Yes

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.