第 23608 題：瑕積分

原題與已知條件

題幹明示integrand、區間、cutoff與每個無界端或奇點的one-sided limit責任。操作=improper-interior-singular-split；資料={"left_scale":8,"right_scale":9,"left_power_numerator":3,"left_power_denominator":4,"right_power_numerator":0,"right_power_denominator":1,"singularity":-5}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。

• G001 — 題幹明示integrand、區間、cutoff與每個無界端或奇點的one-sided limit責任。操作=improper-interior-singular-split；資料={"left_scale":8,"right_scale":9,"left_power_numerator":3,"left_power_denominator":4,"right_power_numerator":0,"right_power_denominator":1,"singularity":-5}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。

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-improper-integral

逐步推導

source operation=improper-interior-singular-split;original snapshot and mathematics retained

explicit ordered source output fields=[interval, singularity, left_integrand, right_integrand, left_integral, right_integral, left_converges, right_converges, sum, certificate]

Improper LEFT reduced exponent numerator gap := sub(4, 3)

Improper LEFT reduced exponent numerator gap = 1

Improper LEFT reciprocal of one minus exponent := div(4, 1)

Improper LEFT reciprocal of one minus exponent = 4

Improper LEFT scale times singular-side integral := mul(8, 4)

Improper LEFT scale times singular-side integral = 32

ORIGINAL LEFT side at c=-5;scale=8;REDUCED exponent=3/4 in[0,1)

put distance t>0 from c;integral_delta^1 scale*t^(-exponent)=scale/(1-exponent)*(1-delta^(1-exponent));positive1-exponent makes delta term vanish,side limit=32

Improper RIGHT reduced exponent numerator gap := sub(1, 0)

Improper RIGHT reduced exponent numerator gap = 1

Improper RIGHT reciprocal of one minus exponent := div(1, 1)

Improper RIGHT reciprocal of one minus exponent = 1

Improper RIGHT scale times singular-side integral := mul(9, 1)

Improper RIGHT scale times singular-side integral = 9

ORIGINAL RIGHT side at c=-5;scale=9;REDUCED exponent=0/1 in[0,1)

put distance t>0 from c;integral_delta^1 scale*t^(-exponent)=scale/(1-exponent)*(1-delta^(1-exponent));positive1-exponent makes delta term vanish,side limit=9

Improper independent singular-side limits sum := sum(32, 9)

Improper independent singular-side limits sum = 41

逐步反向核對

• S010 · premise_and_result_audit — 以精確有理數重算 Improper independent singular-side limits sum，不以小數近似替代。
• S009 · premise_and_result_audit — 回到原始資料與本步條件核對，不把單向推論誤稱等價。
• S008 · premise_and_result_audit — 以精確有理數重算 Improper RIGHT scale times singular-side integral，不以小數近似替代。
• S007 · premise_and_result_audit — 以精確有理數重算 Improper RIGHT reciprocal of one minus exponent，不以小數近似替代。
• S006 · premise_and_result_audit — 以精確有理數重算 Improper RIGHT reduced exponent numerator gap，不以小數近似替代。
• S005 · premise_and_result_audit — 回到原始資料與本步條件核對，不把單向推論誤稱等價。
• S004 · premise_and_result_audit — 以精確有理數重算 Improper LEFT scale times singular-side integral，不以小數近似替代。
• S003 · premise_and_result_audit — 以精確有理數重算 Improper LEFT reciprocal of one minus exponent，不以小數近似替代。
• S002 · premise_and_result_audit — 以精確有理數重算 Improper LEFT reduced exponent numerator gap，不以小數近似替代。
• S001 · premise_and_result_audit — 回到原始資料與本步條件核對，不把單向推論誤稱等價。

結論

([-6,-4], -5, left:8*(c-x)^(-3/4), right:9*(x-c)^(-0/1), 32, 9, 是, 是, 41, interior_singularity_is_split_and_both_one_sided_limits_converge_independently)

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。