{
  "index": "1987-B-1",
  "type": "ANA",
  "tag": [
    "ANA"
  ],
  "difficulty": "",
  "question": "Evaluate\n\\[\n\\int_2^4 \\frac{\\sqrt{\\ln(9-x)}\\,dx}{\\sqrt{\\ln(9-x)}+\\sqrt{\\ln(x+3)}}.\n\\]",
  "solution": "Solution. The integrand is continuous on \\( [2,4] \\). Let \\( I \\) be the value of the integral. As \\( x \\) goes from 2 to \\( 4,9-x \\) and \\( x+3 \\) go from 7 to 5 , and from 5 to 7 , respectively. This symmetry suggests the substitution \\( x=6-y \\) reversing the interval [2,4]. After interchanging the limits of integration, this yields\n\\[\nI=\\int_{2}^{4} \\frac{\\sqrt{\\ln (y+3)} d y}{\\sqrt{\\ln (y+3)}+\\sqrt{\\ln (9-y)}} .\n\\]\n\nThus\n\\[\n2 I=\\int_{2}^{4} \\frac{\\sqrt{\\ln (x+3)}+\\sqrt{\\ln (9-x)}}{\\sqrt{\\ln (x+3)}+\\sqrt{\\ln (9-x)}} d x=\\int_{2}^{4} d x=2,\n\\]\nand \\( I=1 \\).\nRemark. The same argument applies if \\( \\sqrt{\\ln x} \\) is replaced by any continuous function such that \\( f(x+3)+f(9-x) \\neq 0 \\) for \\( 2 \\leq x \\leq 4 \\).",
  "vars": [
    "x",
    "y"
  ],
  "params": [
    "I"
  ],
  "sci_consts": [],
  "variants": {
    "descriptive_long": {
      "map": {
        "x": "variablex",
        "y": "variabley",
        "I": "integralvalue"
      },
      "question": "Evaluate\n\\[\n\\int_2^4 \\frac{\\sqrt{\\ln(9-variablex)}\\,d variablex}{\\sqrt{\\ln(9-variablex)}+\\sqrt{\\ln(variablex+3)}}.\n\\]",
      "solution": "Solution. The integrand is continuous on \\( [2,4] \\). Let \\( integralvalue \\) be the value of the integral. As \\( variablex \\) goes from 2 to \\( 4,9-variablex \\) and \\( variablex+3 \\) go from 7 to 5 , and from 5 to 7 , respectively. This symmetry suggests the substitution \\( variablex=6-variabley \\) reversing the interval [2,4]. After interchanging the limits of integration, this yields\n\\[\nintegralvalue=\\int_{2}^{4} \\frac{\\sqrt{\\ln (variabley+3)} d variabley}{\\sqrt{\\ln (variabley+3)}+\\sqrt{\\ln (9-variabley)}} .\n\\]\n\nThus\n\\[\n2 integralvalue=\\int_{2}^{4} \\frac{\\sqrt{\\ln (variablex+3)}+\\sqrt{\\ln (9-variablex)}}{\\sqrt{\\ln (variablex+3)}+\\sqrt{\\ln (9-variablex)}} d variablex=\\int_{2}^{4} d variablex=2,\n\\]\nand \\( integralvalue=1 \\).\nRemark. The same argument applies if \\( \\sqrt{\\ln variablex} \\) is replaced by any continuous function such that \\( f(variablex+3)+f(9-variablex) \\neq 0 \\) for \\( 2 \\leq variablex \\leq 4 \\)."
    },
    "descriptive_long_confusing": {
      "map": {
        "x": "sapphire",
        "y": "cardinal",
        "I": "harmonica"
      },
      "question": "Evaluate\n\\[\n\\int_2^4 \\frac{\\sqrt{\\ln(9-sapphire)}\\,dsapphire}{\\sqrt{\\ln(9-sapphire)}+\\sqrt{\\ln(sapphire+3)}}.\n\\]",
      "solution": "Solution. The integrand is continuous on \\( [2,4] \\). Let \\( harmonica \\) be the value of the integral. As \\( sapphire \\) goes from 2 to 4, 9-sapphire and sapphire+3 go from 7 to 5 , and from 5 to 7 , respectively. This symmetry suggests the substitution \\( sapphire=6-cardinal \\) reversing the interval [2,4]. After interchanging the limits of integration, this yields\n\\[\nharmonica=\\int_{2}^{4} \\frac{\\sqrt{\\ln (cardinal+3)} d cardinal}{\\sqrt{\\ln (cardinal+3)}+\\sqrt{\\ln (9-cardinal)}} .\n\\]\n\nThus\n\\[\n2 harmonica=\\int_{2}^{4} \\frac{\\sqrt{\\ln (sapphire+3)}+\\sqrt{\\ln (9-sapphire)}}{\\sqrt{\\ln (sapphire+3)}+\\sqrt{\\ln (9-sapphire)}} d sapphire=\\int_{2}^{4} d sapphire=2,\n\\]\nand \\( harmonica=1 \\).\nRemark. The same argument applies if \\( \\sqrt{\\ln sapphire} \\) is replaced by any continuous function such that \\( f(sapphire+3)+f(9-sapphire) \\neq 0 \\) for \\( 2 \\leq sapphire \\leq 4 \\)."
    },
    "descriptive_long_misleading": {
      "map": {
        "x": "constantval",
        "y": "immutableval",
        "I": "derivative"
      },
      "question": "Evaluate\n\\[\n\\int_2^4 \\frac{\\sqrt{\\ln(9-constantval)}\\,d constantval}{\\sqrt{\\ln(9-constantval)}+\\sqrt{\\ln(constantval+3)}}.\n\\]",
      "solution": "Solution. The integrand is continuous on \\( [2,4] \\). Let \\( derivative \\) be the value of the integral. As \\( constantval \\) goes from 2 to \\( 4,9-constantval \\) and \\( constantval+3 \\) go from 7 to 5 , and from 5 to 7 , respectively. This symmetry suggests the substitution \\( constantval=6-immutableval \\) reversing the interval [2,4]. After interchanging the limits of integration, this yields\n\\[\nderivative=\\int_{2}^{4} \\frac{\\sqrt{\\ln (immutableval+3)} d immutableval}{\\sqrt{\\ln (immutableval+3)}+\\sqrt{\\ln (9-immutableval)}} .\n\\]\n\nThus\n\\[\n2 derivative=\\int_{2}^{4} \\frac{\\sqrt{\\ln (constantval+3)}+\\sqrt{\\ln (9-constantval)}}{\\sqrt{\\ln (constantval+3)}+\\sqrt{\\ln (9-constantval)}} d constantval=\\int_{2}^{4} d constantval=2,\n\\]\nand \\( derivative=1 \\).\nRemark. The same argument applies if \\( \\sqrt{\\ln constantval} \\) is replaced by any continuous function such that \\( f(constantval+3)+f(9-constantval) \\neq 0 \\) for \\( 2 \\leq constantval \\leq 4 \\)."
    },
    "garbled_string": {
      "map": {
        "x": "mifrntlz",
        "y": "gkdlzpra",
        "I": "qzxwvtnp"
      },
      "question": "Evaluate\n\\[\n\\int_2^4 \\frac{\\sqrt{\\ln(9-mifrntlz)}\\,d mifrntlz}{\\sqrt{\\ln(9-mifrntlz)}+\\sqrt{\\ln(mifrntlz+3)}}.\n\\]",
      "solution": "Solution. The integrand is continuous on \\( [2,4] \\). Let \\( qzxwvtnp \\) be the value of the integral. As \\( mifrntlz \\) goes from 2 to \\( 4,9-mifrntlz \\) and \\( mifrntlz+3 \\) go from 7 to 5 , and from 5 to 7 , respectively. This symmetry suggests the substitution \\( mifrntlz=6-gkdlzpra \\) reversing the interval [2,4]. After interchanging the limits of integration, this yields\n\\[\nqzxwvtnp=\\int_{2}^{4} \\frac{\\sqrt{\\ln (gkdlzpra+3)} d gkdlzpra}{\\sqrt{\\ln (gkdlzpra+3)}+\\sqrt{\\ln (9-gkdlzpra)}} .\n\\]\n\nThus\n\\[\n2 qzxwvtnp=\\int_{2}^{4} \\frac{\\sqrt{\\ln (mifrntlz+3)}+\\sqrt{\\ln (9-mifrntlz)}}{\\sqrt{\\ln (mifrntlz+3)}+\\sqrt{\\ln (9-mifrntlz)}} d mifrntlz=\\int_{2}^{4} d mifrntlz=2,\n\\]\nand \\( qzxwvtnp=1 \\).\nRemark. The same argument applies if \\( \\sqrt{\\ln(mifrntlz)} \\) is replaced by any continuous function such that \\( f(mifrntlz+3)+f(9-mifrntlz) \\neq 0 \\) for \\( 2 \\leq mifrntlz \\leq 4 \\)."
    },
    "kernel_variant": {
      "question": "Let\n\n  G(x₁,…,x₇)=⎛⎜⎜⎝ ∑_{k=1}^{7} Li_{7/2}\\!\\Bigl[\\arctan^{2}\\!\\bigl(√{\\,13-x_k\\,}\\bigr)\\Bigr] ⎞⎟⎟⎠^{\\!1/3},\n\nwhere Li_{7/2}(z)=∑_{m=1}^{∞} z^{m}/m^{7/2} is the polylogarithm of order 7⁄2.  \nEvaluate the seven–fold integral  \n\n  I = ∫_{0}^{13} … ∫_{0}^{13}  \n         G(x₁,…,x₇)  \n   ──────────────────────────────────────── d x₁⋯d x₇ ,\n         G(x₁,…,x₇)+G(13−x₁,…,13−x₇)  \n\nthat is, find the exact value of  \n\n  I = ∬_{[0,13]^7}  \n         G(𝐱)  \n   ──────────────────────── d^{7}𝐱 ,\n         G(𝐱)+G(13−𝐱)\n\n",
      "solution": "We reproduce the style of the original one–variable argument, but now in seven dimensions.\n\nStep 1 (hypercube symmetry).  \nDenote the closed seven–dimensional cube  \n\n  Q=[0,13]^7 ⊂ℝ⁷,\n\nand write the integrand as  \n\n  f(𝐱)= G(𝐱) / [ G(𝐱)+G(𝐜−𝐱) ],  𝐜:=(13,…,13).\n\nBecause every component of 𝐜−𝐱 lies again in [0,13], the map  \n\n  T:Q→Q, T(𝐱)=𝐜−𝐱\n\nis a bijection of the domain onto itself.  Moreover  \n\n  f(T(𝐱))= G(𝐜−𝐱) / [ G(𝐜−𝐱)+G(𝐱) ]\n       = 1 − f(𝐱).                                                 (1)\n\nStep 2 (change of variables).  \nApply T to the integral:\n\n  I = ∫_{Q} f(𝐱) d^{7}𝐱\n    = ∫_{Q} f(T(𝐱)) |det DT| d^{7}𝐱   (Jacobian = 1, because T is a translation+reflection)\n    = ∫_{Q} [1−f(𝐱)] d^{7}𝐱   by (1).\n\nStep 3 (adding the two faces of I).  \nAdding the original form of I to the transformed one we obtain\n\n  2I = ∫_{Q} [f(𝐱)+1−f(𝐱)] d^{7}𝐱 = ∫_{Q} 1 d^{7}𝐱 = Vol(Q).            (2)\n\nStep 4 (volume of the 7–cube).  \nEach edge of Q has length 13; hence  \n\n  Vol(Q)=13^{7}=62 748 517.\n\nInsert this into (2):\n\n  2I = 13^{7} ⇒ I = 13^{7}/2 = 62 748 517 ⁄ 2.\n\nTherefore  \n\n  boxed I = 13^{7}/2.\n\n",
      "_replacement_note": {
        "replaced_at": "2025-07-05T22:17:12.112547",
        "reason": "Original kernel variant was too easy compared to the original problem"
      }
    }
  },
  "checked": true,
  "problem_type": "calculation"
}