第 23695 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-two-sided-independent；資料={"left_scale":20,"right_scale":21,"exponent":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-two-sided-independent；資料={"left_scale":20,"right_scale":21,"exponent":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-two-sided-independent;original snapshot and mathematics retained

explicit ordered source output fields=[piecewise_integrand, interval, left_tail_integral, right_tail_integral, left_converges, right_converges, sum, certificate]

Improper ORIGINAL negative tail after x=-u lower positive power step1 := mul(1, 1)

Improper ORIGINAL negative tail after x=-u lower positive power step1 = 1

Improper ORIGINAL negative tail after x=-u lower positive power step2 := mul(1, 1)

Improper ORIGINAL negative tail after x=-u lower positive power step2 = 1

Improper ORIGINAL negative tail after x=-u lower positive power step3 := mul(1, 1)

Improper ORIGINAL negative tail after x=-u lower positive power step3 = 1

Improper ORIGINAL negative tail after x=-u lower positive power step4 := mul(1, 1)

Improper ORIGINAL negative tail after x=-u lower positive power step4 = 1

Improper ORIGINAL negative tail after x=-u exponent-minus-one times lower-power := mul(4, 1)

Improper ORIGINAL negative tail after x=-u exponent-minus-one times lower-power = 4

Improper ORIGINAL negative tail after x=-u exact antiderivative limit value := div(20, 4)

Improper ORIGINAL negative tail after x=-u exact antiderivative limit value = 5

Improper ORIGINAL negative tail after x=-u:ORIGINAL scale=20,exponent=5>1,lower=1>=1;integrand scale/x^exponent

for EVERY B>=lower,integral_lower^B=scale/(exponent-1)*(lower^(1-exponent)-B^(1-exponent));B^(1-exponent) tends0,so limit=5

Improper ORIGINAL positive tail lower positive power step1 := mul(1, 1)

Improper ORIGINAL positive tail lower positive power step1 = 1

Improper ORIGINAL positive tail lower positive power step2 := mul(1, 1)

Improper ORIGINAL positive tail lower positive power step2 = 1

Improper ORIGINAL positive tail lower positive power step3 := mul(1, 1)

Improper ORIGINAL positive tail lower positive power step3 = 1

Improper ORIGINAL positive tail lower positive power step4 := mul(1, 1)

Improper ORIGINAL positive tail lower positive power step4 = 1

Improper ORIGINAL positive tail exponent-minus-one times lower-power := mul(4, 1)

Improper ORIGINAL positive tail exponent-minus-one times lower-power = 4

Improper ORIGINAL positive tail exact antiderivative limit value := div(21, 4)

Improper ORIGINAL positive tail exact antiderivative limit value = 21/4

Improper ORIGINAL positive tail:ORIGINAL scale=21,exponent=5>1,lower=1>=1;integrand scale/x^exponent

for EVERY B>=lower,integral_lower^B=scale/(exponent-1)*(lower^(1-exponent)-B^(1-exponent));B^(1-exponent) tends0,so limit=21/4

Improper independently convergent tails plus zero middle := sum(5, 21/4)

Improper independently convergent tails plus zero middle = 41/4

ORIGINAL piecewise scales left=20,right=21,exponent=5;middle(-1,1) integrand0

negative-tail limit=5;positive-tail limit=21/4;BOTH independently finite,then ordinary sum=41/4

逐步反向核對

• S017 · 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。
• S016 · premise_and_result_audit — 以精確有理數重算 Improper independently convergent tails plus zero middle，不以小數近似替代。
• S015 · 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。
• S014 · premise_and_result_audit — 以精確有理數重算 Improper ORIGINAL positive tail exact antiderivative limit value，不以小數近似替代。
• S013 · premise_and_result_audit — 以精確有理數重算 Improper ORIGINAL positive tail exponent-minus-one times lower-power，不以小數近似替代。
• S012 · premise_and_result_audit — 以精確有理數重算 Improper ORIGINAL positive tail lower positive power step4，不以小數近似替代。
• S011 · premise_and_result_audit — 以精確有理數重算 Improper ORIGINAL positive tail lower positive power step3，不以小數近似替代。
• S010 · premise_and_result_audit — 以精確有理數重算 Improper ORIGINAL positive tail lower positive power step2，不以小數近似替代。
• S009 · premise_and_result_audit — 以精確有理數重算 Improper ORIGINAL positive tail lower positive power step1，不以小數近似替代。
• S008 · 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。
• S007 · premise_and_result_audit — 以精確有理數重算 Improper ORIGINAL negative tail after x=-u exact antiderivative limit value，不以小數近似替代。
• S006 · premise_and_result_audit — 以精確有理數重算 Improper ORIGINAL negative tail after x=-u exponent-minus-one times lower-power，不以小數近似替代。
• S005 · premise_and_result_audit — 以精確有理數重算 Improper ORIGINAL negative tail after x=-u lower positive power step4，不以小數近似替代。
• S004 · premise_and_result_audit — 以精確有理數重算 Improper ORIGINAL negative tail after x=-u lower positive power step3，不以小數近似替代。
• S003 · premise_and_result_audit — 以精確有理數重算 Improper ORIGINAL negative tail after x=-u lower positive power step2，不以小數近似替代。
• S002 · premise_and_result_audit — 以精確有理數重算 Improper ORIGINAL negative tail after x=-u lower positive power step1，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

(x<=-1:20/abs(x)^5;-1<x<1:0;x>=1:21/x^5, (-infinity,infinity), 5, 21/4, 是, 是, 41/4, both_one_sided_tail_integrals_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。