Question 1396: Antiderivative Value Retaining the Actual Constant

Original question and givens

On all R, f has ascending coefficients [8, 21, 1, 1], starting with its constant term. Let F0 be its zero-constant primitive. Find F0(3)+(8), retaining the actual additive constant. Give a reduced fraction with positive denominator, retaining /1 for integers.

• G001 — On all R, f has ascending coefficients [8, 21, 1, 1], starting with its constant term. Let F0 be its zero-constant primitive. Find F0(3)+(8), retaining the actual additive constant. Give a reduced fraction with positive denominator, retaining /1 for integers.

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]

f coefficients ascending=[8, 21, 1, 1]

seek F₀ with constant coefficient0 and F₀′=f

primitive coefficient at degree1 := div(8, 1)

primitive coefficient at degree1 = 8

differentiate back coefficient0 := mul(8, 1)

differentiate back coefficient0 = 8

primitive coefficient at degree2 := div(21, 2)

primitive coefficient at degree2 = 21/2

differentiate back coefficient1 := mul(21/2, 2)

differentiate back coefficient1 = 21

primitive coefficient at degree3 := div(1, 3)

primitive coefficient at degree3 = 1/3

differentiate back coefficient2 := mul(1/3, 3)

differentiate back coefficient2 = 1

primitive coefficient at degree4 := div(1, 4)

primitive coefficient at degree4 = 1/4

differentiate back coefficient3 := mul(1/4, 4)

differentiate back coefficient3 = 1

[0, 8, 21/2, 1/3, 1/4]

all derivative coefficients agree with f

constant coefficient+C := sum(0, 8)

constant coefficient+C = 8

F(x) multiply at degree4 := mul(0, 3)

F(x) multiply at degree4 = 0

F(x) add coefficient4 := sum(0, 1/4)

F(x) add coefficient4 = 1/4

F(x) multiply at degree3 := mul(1/4, 3)

F(x) multiply at degree3 = 3/4

F(x) add coefficient3 := sum(3/4, 1/3)

F(x) add coefficient3 = 13/12

F(x) multiply at degree2 := mul(13/12, 3)

F(x) multiply at degree2 = 13/4

F(x) add coefficient2 := sum(13/4, 21/2)

F(x) add coefficient2 = 55/4

F(x) multiply at degree1 := mul(55/4, 3)

F(x) multiply at degree1 = 165/4

F(x) add coefficient1 := sum(165/4, 8)

F(x) add coefficient1 = 197/4

F(x) multiply at degree0 := mul(197/4, 3)

F(x) multiply at degree0 = 591/4

F(x) add coefficient0 := sum(591/4, 8)

F(x) add coefficient0 = 623/4

x=3; additive constant=8

F(x)=623/4

Reverse checks

• S023 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S022 · premise_and_result_audit — Recompute F(x) add coefficient0 with exact rational arithmetic, not rounded decimals.
• S021 · premise_and_result_audit — Recompute F(x) multiply at degree0 with exact rational arithmetic, not rounded decimals.
• S020 · premise_and_result_audit — Recompute F(x) add coefficient1 with exact rational arithmetic, not rounded decimals.
• S019 · premise_and_result_audit — Recompute F(x) multiply at degree1 with exact rational arithmetic, not rounded decimals.
• S018 · premise_and_result_audit — Recompute F(x) add coefficient2 with exact rational arithmetic, not rounded decimals.
• S017 · premise_and_result_audit — Recompute F(x) multiply at degree2 with exact rational arithmetic, not rounded decimals.
• S016 · premise_and_result_audit — Recompute F(x) add coefficient3 with exact rational arithmetic, not rounded decimals.
• S015 · premise_and_result_audit — Recompute F(x) multiply at degree3 with exact rational arithmetic, not rounded decimals.
• S014 · premise_and_result_audit — Recompute F(x) add coefficient4 with exact rational arithmetic, not rounded decimals.
• S013 · premise_and_result_audit — Recompute F(x) multiply at degree4 with exact rational arithmetic, not rounded decimals.
• S012 · premise_and_result_audit — Recompute constant coefficient+C with exact rational arithmetic, not rounded decimals.
• S011 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• 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

623/4

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.