第 32656 題：單項式的有向矩積分

原題與已知條件

全實數論域，以下函數／點均以明示實數模型解讀。f(x)=-6x^5，n為0至5。依明示多項式矩公式求從 a=-3 到 b=0 的有向積分，採最簡正分母分數。

• G001 — 全實數論域，以下函數／點均以明示實數模型解讀。f(x)=-6x^5，n為0至5。依明示多項式矩公式求從 a=-3 到 b=0 的有向積分，採最簡正分母分數。

定義與適用範圍

沿用原題明示的論域、假設與來源 MKO 定義。本題已完成第二對話數理與中英語義覆核，並由原始參數逐題獨立重算模型、答案與每個算術步驟；原題仍為 candidate。

• mko-definite-integral

逐步推導

source roles = [qf-integral-monomial, monomial_integral, reals, []]

derived = [[signed_moment_fraction_RETAIN1]]

Monomial ORIGINAL term lower endpoint ORIGINAL power multiplication1 := mul(1, -3)

Monomial ORIGINAL term lower endpoint ORIGINAL power multiplication1 = -3

Monomial ORIGINAL term lower endpoint ORIGINAL power multiplication2 := mul(-3, -3)

Monomial ORIGINAL term lower endpoint ORIGINAL power multiplication2 = 9

Monomial ORIGINAL term lower endpoint ORIGINAL power multiplication3 := mul(9, -3)

Monomial ORIGINAL term lower endpoint ORIGINAL power multiplication3 = -27

Monomial ORIGINAL term lower endpoint ORIGINAL power multiplication4 := mul(-27, -3)

Monomial ORIGINAL term lower endpoint ORIGINAL power multiplication4 = 81

Monomial ORIGINAL term lower endpoint ORIGINAL power multiplication5 := mul(81, -3)

Monomial ORIGINAL term lower endpoint ORIGINAL power multiplication5 = -243

Monomial ORIGINAL term lower endpoint ORIGINAL power multiplication6 := mul(-243, -3)

Monomial ORIGINAL term lower endpoint ORIGINAL power multiplication6 = 729

source roles = [Monomial ORIGINAL term lower endpoint, -3, 6]

derived = [729]

Monomial ORIGINAL term upper endpoint ORIGINAL power multiplication1 := mul(1, 0)

Monomial ORIGINAL term upper endpoint ORIGINAL power multiplication1 = 0

Monomial ORIGINAL term upper endpoint ORIGINAL power multiplication2 := mul(0, 0)

Monomial ORIGINAL term upper endpoint ORIGINAL power multiplication2 = 0

Monomial ORIGINAL term upper endpoint ORIGINAL power multiplication3 := mul(0, 0)

Monomial ORIGINAL term upper endpoint ORIGINAL power multiplication3 = 0

Monomial ORIGINAL term upper endpoint ORIGINAL power multiplication4 := mul(0, 0)

Monomial ORIGINAL term upper endpoint ORIGINAL power multiplication4 = 0

Monomial ORIGINAL term upper endpoint ORIGINAL power multiplication5 := mul(0, 0)

Monomial ORIGINAL term upper endpoint ORIGINAL power multiplication5 = 0

Monomial ORIGINAL term upper endpoint ORIGINAL power multiplication6 := mul(0, 0)

Monomial ORIGINAL term upper endpoint ORIGINAL power multiplication6 = 0

source roles = [Monomial ORIGINAL term upper endpoint, 0, 6]

derived = [0]

Monomial ORIGINAL term ORIGINAL upper power minus lower power := sub(0, 729)

Monomial ORIGINAL term ORIGINAL upper power minus lower power = -729

Monomial ORIGINAL term ORIGINAL coefficient times power difference := mul(-6, -729)

Monomial ORIGINAL term ORIGINAL coefficient times power difference = 4374

Monomial ORIGINAL term ORIGINAL signed moment over positive exponent := div(4374, 6)

Monomial ORIGINAL term ORIGINAL signed moment over positive exponent = 729

source roles = [Monomial ORIGINAL term, -6, 5, -3, 0]

derived = [729, 0, -729, 729/1]

逐步反向核對

• S019 · premise_and_result_audit — 以原始資料、欄位角色、型別、論域及本步前提反向代回核對；不表示結論能唯一還原前提。來源綁定候選、獨立審查、形式證明與發布分開。
• S018 · premise_and_result_audit — 以精確有理數重算 Monomial ORIGINAL term ORIGINAL signed moment over positive exponent，不以小數近似替代。
• S017 · premise_and_result_audit — 以精確有理數重算 Monomial ORIGINAL term ORIGINAL coefficient times power difference，不以小數近似替代。
• S016 · premise_and_result_audit — 以精確有理數重算 Monomial ORIGINAL term ORIGINAL upper power minus lower power，不以小數近似替代。
• S015 · premise_and_result_audit — 以原始資料、欄位角色、型別、論域及本步前提反向代回核對；不表示結論能唯一還原前提。來源綁定候選、獨立審查、形式證明與發布分開。
• S014 · premise_and_result_audit — 以精確有理數重算 Monomial ORIGINAL term upper endpoint ORIGINAL power multiplication6，不以小數近似替代。
• S013 · premise_and_result_audit — 以精確有理數重算 Monomial ORIGINAL term upper endpoint ORIGINAL power multiplication5，不以小數近似替代。
• S012 · premise_and_result_audit — 以精確有理數重算 Monomial ORIGINAL term upper endpoint ORIGINAL power multiplication4，不以小數近似替代。
• S011 · premise_and_result_audit — 以精確有理數重算 Monomial ORIGINAL term upper endpoint ORIGINAL power multiplication3，不以小數近似替代。
• S010 · premise_and_result_audit — 以精確有理數重算 Monomial ORIGINAL term upper endpoint ORIGINAL power multiplication2，不以小數近似替代。
• S009 · premise_and_result_audit — 以精確有理數重算 Monomial ORIGINAL term upper endpoint ORIGINAL power multiplication1，不以小數近似替代。
• S008 · premise_and_result_audit — 以原始資料、欄位角色、型別、論域及本步前提反向代回核對；不表示結論能唯一還原前提。來源綁定候選、獨立審查、形式證明與發布分開。
• S007 · premise_and_result_audit — 以精確有理數重算 Monomial ORIGINAL term lower endpoint ORIGINAL power multiplication6，不以小數近似替代。
• S006 · premise_and_result_audit — 以精確有理數重算 Monomial ORIGINAL term lower endpoint ORIGINAL power multiplication5，不以小數近似替代。
• S005 · premise_and_result_audit — 以精確有理數重算 Monomial ORIGINAL term lower endpoint ORIGINAL power multiplication4，不以小數近似替代。
• S004 · premise_and_result_audit — 以精確有理數重算 Monomial ORIGINAL term lower endpoint ORIGINAL power multiplication3，不以小數近似替代。
• S003 · premise_and_result_audit — 以精確有理數重算 Monomial ORIGINAL term lower endpoint ORIGINAL power multiplication2，不以小數近似替代。
• S002 · premise_and_result_audit — 以精確有理數重算 Monomial ORIGINAL term lower endpoint ORIGINAL power multiplication1，不以小數近似替代。
• S001 · premise_and_result_audit — 以原始資料、欄位角色、型別、論域及本步前提反向代回核對；不表示結論能唯一還原前提。來源綁定候選、獨立審查、形式證明與發布分開。

結論

729/1

依上列完整代值及條件核對得到原題答案；原題編號、格式與狀態保持不變。

本題覆核與來源綁定

覆核日期：2026-10-26

原題、答案、步驟表達式、數值 AST 與既有歷史保持原樣；原來的候選狀態屬於來源歷史，不覆寫為形式化證明。