{
  "schema_version": "ocme-derivation-reader-v1",
  "guide_id": "derivation-mko-improper-integral-01",
  "mko_id": "mko-improper-integral",
  "title": {
    "zh-Hant": "每個無界端與奇點分別取極限",
    "en": "Separate limits at every infinite endpoint and singularity"
  },
  "contract": {
    "schema_version": "ocme-derivation-guide-v0.1",
    "guide_id": "derivation-mko-improper-integral-01",
    "source": {
      "mko_id": "mko-improper-integral",
      "commit": "b1b2f1dcf8c50d67a93fe52d95f8eef7b3abba9f",
      "sha256": "db7f78b411add242ee46cc38c5fede57caf33f99eeb955e7bbe6271da2d44dec",
      "english_sha256": "4395ebe9f67586ef48cc68baaec03b6a479aa7a30603cacff5d2740e4d57b731"
    },
    "mode": "worked_example",
    "audience_prerequisites": [
      "algebra",
      "epsilon",
      "FTC",
      "definition_here",
      "log_integral_boundary"
    ],
    "goal": {
      "zh-Hant": "計算三種明示冪函數瑕積分，並區分 principal value。",
      "en": "Compute three explicit power-law improper integrals and distinguish principal value."
    },
    "givens": [
      {
        "id": "G1",
        "statement": {
          "zh-Hant": "一側 f(x)=3/x^2，x≥2；兩側 g(x)=2/|x|^2（x≤-1）、0（-1<x<1）、3/x^2（x≥1）。",
          "en": "One side: f(x)=3/x^2 for x≥2. Two sides: g(x)=2/|x|^2 for x≤-1, zero for -1<x<1, and 3/x^2 for x≥1."
        }
      },
      {
        "id": "G2",
        "statement": {
          "zh-Hant": "奇點例 h(x)=(-x)^(-1/2)（-1≤x<0）、2x^(-1/2)（0<x≤1），於 0 分拆。",
          "en": "Singularity example: h(x)=(-x)^(-1/2) for -1≤x<0 and 2*x^(-1/2) for 0<x≤1, split at zero."
        }
      }
    ],
    "steps": [
      {
        "id": "S01",
        "uses": [
          "G1"
        ],
        "before_tex": "b>2",
        "after_tex": "\\int_2^b3x^{-2}\\,dx=[-3/x]_2^b=3/2-3/b",
        "relation": "implies",
        "rule": "FTC",
        "conditions": [
          "b>2"
        ],
        "explanation": {
          "zh-Hant": "先只做合法有限區間積分，未直接操作無窮端點。",
          "en": "First integrate on a qualified finite interval, without directly substituting an infinite endpoint."
        },
        "reverse_audit": {
          "kind": "premise_audit",
          "required_ids": [
            "G1"
          ],
          "explanation": {
            "zh-Hant": "b>2 使 integrand 連續且分母正。",
            "en": "b>2 keeps the integrand continuous and denominators positive."
          }
        },
        "leaf_basis": "FTC"
      },
      {
        "id": "S02",
        "uses": [
          "S01"
        ],
        "before_tex": "3/b\\to0\\quad(b\\to\\infty)",
        "after_tex": "\\int_2^\\infty3x^{-2}\\,dx=3/2",
        "relation": "definition",
        "rule": "definition_here",
        "conditions": [],
        "explanation": {
          "zh-Hant": "任意 epsilon 取 b>3/epsilon 即嚴格控尾，按定義得一側瑕積分。",
          "en": "For any epsilon, b>3/epsilon controls the entire tail strictly, defining the one-sided integral."
        },
        "reverse_audit": {
          "kind": "premise_audit",
          "required_ids": [
            "S01"
          ],
          "explanation": {
            "zh-Hant": "不是只算若干 b 值見其接近。",
            "en": "This is not a conclusion from several close cutoff values."
          }
        },
        "leaf_basis": "definition_here"
      },
      {
        "id": "S03",
        "uses": [
          "G1"
        ],
        "before_tex": "A>1,\\ B>1",
        "after_tex": "\\int_{-A}^{-1}g=2-2/A,\\quad\\int_1^B g=3-3/B",
        "relation": "implies",
        "rule": "FTC",
        "conditions": [],
        "explanation": {
          "zh-Hant": "左邊以 u=-x 代換，右邊直接算；兩個 cutoff 獨立。",
          "en": "Use u=-x on the left and integrate directly on the right; the two cutoffs are independent."
        },
        "reverse_audit": {
          "kind": "premise_audit",
          "required_ids": [
            "G1"
          ],
          "explanation": {
            "zh-Hant": "不能綁 A=B 才取得通常瑕積分資格。",
            "en": "Qualification must not rely on tying A and B together."
          }
        },
        "leaf_basis": "FTC"
      },
      {
        "id": "S04",
        "uses": [
          "S03"
        ],
        "before_tex": "A\\to\\infty\\text{ independently of }B\\to\\infty",
        "after_tex": "\\int_{-\\infty}^{\\infty}g=2+0+3=5",
        "relation": "definition",
        "rule": "definition_here",
        "conditions": [],
        "explanation": {
          "zh-Hant": "左右尾各自存在，中段為零才可加總。",
          "en": "Both tails exist separately, and only then are they added to the zero middle part."
        },
        "reverse_audit": {
          "kind": "premise_audit",
          "required_ids": [
            "S03"
          ],
          "explanation": {
            "zh-Hant": "對稱抵銷沒有參與，兩側責任獨立完成。",
            "en": "There is no symmetric cancellation; each side is independently qualified."
          }
        },
        "leaf_basis": "definition_here"
      },
      {
        "id": "S05",
        "uses": [
          "G2"
        ],
        "before_tex": "0<\\delta<1",
        "after_tex": "\\int_{-1}^{-\\delta}h=2-2\\sqrt\\delta,\\quad\\int_{\\delta}^{1}h=4-4\\sqrt\\delta",
        "relation": "implies",
        "rule": "FTC",
        "conditions": [
          "0<\\delta<1"
        ],
        "explanation": {
          "zh-Hant": "奇點左右各有自己的有限積分式；左側用 u=-x，右側有 scale 2。",
          "en": "Each side of the singularity has its own finite integral; substitute u=-x on the left and retain scale 2 on the right."
        },
        "reverse_audit": {
          "kind": "premise_audit",
          "required_ids": [
            "G2"
          ],
          "explanation": {
            "zh-Hant": "同寫 delta 只是記號便利，兩側極限仍須分別存在。",
            "en": "Using one symbol delta is convenient notation; the two one-sided limits must still exist separately."
          }
        },
        "leaf_basis": "FTC"
      },
      {
        "id": "S06",
        "uses": [
          "S05"
        ],
        "before_tex": "\\sqrt\\delta\\to0\\quad(\\delta\\downarrow0)",
        "after_tex": "\\int_{-1}^{1}h=2+4=6",
        "relation": "definition",
        "rule": "definition_here",
        "conditions": [],
        "explanation": {
          "zh-Hant": "sqrt(delta)<epsilon 可由 delta<epsilon^2 確保，兩側均有限後相加。",
          "en": "The condition delta<epsilon^2 makes sqrt(delta)<epsilon; add the two finite one-sided limits."
        },
        "reverse_audit": {
          "kind": "premise_audit",
          "required_ids": [
            "S05"
          ],
          "explanation": {
            "zh-Hant": "未把只在 0 的缺值當阻礙；真正責任是兩側可積極限。",
            "en": "A missing value at zero is not the obstruction; the one-sided integrability limits are the obligations."
          }
        },
        "leaf_basis": "definition_here"
      },
      {
        "id": "S07",
        "uses": [
          "G1",
          "log_integral_boundary"
        ],
        "before_tex": "j(x)=1/x\\quad(|x|\\ge1),\\qquad j(x)=0\\quad(|x|<1)",
        "after_tex": "\\int_{-A}^{-1}j=-\\log A,\\quad\\int_1^B j=\\log B",
        "relation": "implies",
        "rule": "log_integral_boundary",
        "conditions": [
          "(\\log|x|)'=1/x,\\quad\\log B\\to\\infty"
        ],
        "explanation": {
          "zh-Hant": "此對照函數的兩尾各發散；導數 log|x|=1/x 作為本反例的有限積分依據。",
          "en": "In this contrast example the two tails diverge; the derivative of log|x| is 1/x, supplying its finite antiderivative."
        },
        "reverse_audit": {
          "kind": "premise_audit",
          "required_ids": [
            "G1",
            "log_integral_boundary"
          ],
          "explanation": {
            "zh-Hant": "此處只用已明示的 log 導數反例前提，未把它當新 exp/log 理論。",
            "en": "Only the stated logarithmic-antiderivative premise for this counterexample is used, not a new exp/log theory."
          }
        },
        "leaf_basis": "log_integral_boundary"
      },
      {
        "id": "S08",
        "uses": [
          "S07"
        ],
        "before_tex": "A=B\\Longrightarrow-\\log A+\\log A=0",
        "after_tex": "\\operatorname{PV}\\int j=0\\text{ but the ordinary improper integral does not exist}",
        "relation": "implies",
        "rule": "algebra",
        "conditions": [],
        "explanation": {
          "zh-Hant": "對稱 principal value 的零不補救兩尾發散，所以不能冒充通常瑕積分。",
          "en": "A zero symmetric principal value does not repair divergent individual tails and is not an ordinary improper integral."
        },
        "reverse_audit": {
          "kind": "premise_audit",
          "required_ids": [
            "S07"
          ],
          "explanation": {
            "zh-Hant": "只否定通常積分，並未否定此對稱極限的確為零。",
            "en": "The ordinary integral is rejected without denying the symmetric limit is zero."
          }
        },
        "leaf_basis": "algebra"
      }
    ],
    "result": {
      "zh-Hant": "三個合格瑕積分分別為 3/2、5、6；對照 j 只有對稱主值 0。",
      "en": "The three qualified improper integrals are 3/2, 5 and 6; contrast j has only symmetric principal value zero."
    },
    "negative_controls": [
      {
        "claim": {
          "zh-Hant": "左右發散但抵銷就有通常積分。",
          "en": "Divergent sides cancel into an ordinary integral."
        },
        "expected": "reject",
        "reason": {
          "zh-Hant": "j 兩個一側極限都不有限。",
          "en": "Neither one-sided limit of j is finite."
        }
      },
      {
        "claim": {
          "zh-Hant": "奇點左右只需一側收斂。",
          "en": "Only one side of an interior singularity must converge."
        },
        "expected": "reject",
        "reason": {
          "zh-Hant": "通常瑕積分要求每個奇點兩側分別存在。",
          "en": "An ordinary improper integral requires both one-sided limits at each singularity."
        }
      },
      {
        "claim": {
          "zh-Hant": "幾個大 cutoff 接近即證收斂。",
          "en": "Several close large-cutoff values prove convergence."
        },
        "expected": "reject",
        "reason": {
          "zh-Hant": "未給所有更遠 cutoff 的誤差界。",
          "en": "No bound for every later cutoff is supplied."
        }
      }
    ],
    "status": "candidate",
    "gaps": [],
    "verification": {
      "structure": "checked_local_script",
      "mathematics": "candidate_argument",
      "formalization": "not_formalized"
    },
    "rule_glossary": [
      {
        "id": "mko_definition",
        "kind": "cited",
        "statement": {
          "zh-Hant": "一側無界積分int_a^infinity f由b→infinity的有限區間積分極限定義；本輪明示f(x)=scale/x^p於[lower,infinity)。兩側無界exact family明示f(x)=left_scale/|x|^p於x≤-1、f(x)=0於-1<x<1、f(x)=right_scale/x^p於x≥1，左右tails必須獨立收斂。內部奇點family在[c-1,c+1]明示左右piecewise powers並於c左右分拆，兩側皆收斂。不能只靠symmetric principal value；absolute convergence充分但非必要，有限cutoffs看似穩定不足以證收斂。",
          "en": "A one-sided integral int_a^infinity f, from a to infinity, is the limit of finite-cutoff integrals as b→infinity; this slice declares f(x)=scale/x^p on [lower,infinity). The two-sided exact family declares f(x)=left_scale/|x|^p on x≤−1, f(x)=0 on −1<x<1 and f(x)=right_scale/x^p on x≥1; its tails must converge separately. An interior-singularity family explicitly gives both power-law pieces on [c−1,c+1] and splits at c. A symmetric principal value is insufficient. Absolute convergence is sufficient but not necessary, and apparently stable finite cutoffs do not prove convergence."
        },
        "source": {
          "mko_id": "mko-improper-integral",
          "sha256": "db7f78b411add242ee46cc38c5fede57caf33f99eeb955e7bbe6271da2d44dec",
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        }
      },
      {
        "id": "algebra",
        "kind": "prerequisite",
        "statement": {
          "zh-Hant": "實數域的等式運算；除法前分母必非零，不等式乘正數保持方向。",
          "en": "Use real-field identities; division requires a nonzero denominator and multiplication by a positive number preserves inequalities."
        }
      },
      {
        "id": "epsilon",
        "kind": "prerequisite",
        "statement": {
          "zh-Hant": "收斂表示每個正 epsilon 有一個整數 N，使全部 n≥N 的誤差嚴格小於 epsilon；使用實數 Archimedean 性。",
          "en": "Convergence requires for every positive epsilon an integer N controlling every n≥N by a strict error bound; use the Archimedean property of the reals."
        }
      },
      {
        "id": "FTC",
        "kind": "prerequisite",
        "statement": {
          "zh-Hant": "有限閉區間上連續冪函數可用微積分基本定理；(-c/x)'=c/x^2，(2sqrt(x))'=x^(-1/2)（x>0）。",
          "en": "For continuous power laws on finite closed intervals use the fundamental theorem of calculus; the derivative of -c/x is c/x^2, and that of 2sqrt(x) is x^(-1/2) for x>0."
        }
      },
      {
        "id": "definition_here",
        "kind": "prerequisite",
        "statement": {
          "zh-Hant": "通常瑕積分要求每個無界端或內部奇點的每一側有限截斷積分各有有限極限，再相加；principal value 是另行綁定對稱 cutoff 的極限。",
          "en": "An ordinary improper integral requires a finite limit of finite-cutoff integrals separately at each infinite endpoint and each side of an interior singularity, then adds them; principal value separately binds symmetric cutoffs."
        }
      },
      {
        "id": "log_integral_boundary",
        "kind": "prerequisite",
        "statement": {
          "zh-Hant": "在非零實數上 log|x| 導數為 1/x，且 log B 隨 B→infinity 發散至 infinity。",
          "en": "For nonzero reals the derivative of log|x| is 1/x, and log B tends to infinity as B tends to infinity."
        }
      }
    ],
    "revision_log": [],
    "numeric_checks": [
      {
        "id": "finite-cutoff",
        "lhs": "3/2-3/10",
        "rhs": "6/5",
        "step": "S01"
      }
    ]
  },
  "presentation": {
    "md_scope": {
      "zh-Hant": "只算完整 piecewise 冪函數；不由有限 cutoff 證一般可積性。",
      "en": "Only complete piecewise power laws, not general integrability from finite cutoffs."
    },
    "md_symbols": [
      [
        "A,B,b",
        "獨立無界 cutoff",
        "independent infinite-end cutoffs"
      ],
      [
        "\\delta",
        "奇點一側 cutoff",
        "one-sided singular cutoff"
      ],
      [
        "\\operatorname{PV}",
        "對稱主值",
        "symmetric principal value"
      ]
    ],
    "md_error_controls": [
      {
        "title": "每端分帳 / Separate endpoint accounts",
        "zh": "無界端及奇點左右分別記錄極限。",
        "en": "Record each infinite endpoint and singularity side separately."
      },
      {
        "title": "主值分層 / Principal-value separation",
        "zh": "對稱 cutoff 只證 principal value，不證獨立兩尾。",
        "en": "Symmetric cutoffs certify principal value, not independent tails."
      }
    ],
    "md_open_items": []
  },
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