设函数

原始 JSON

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      "A short note: 设函数\n \nf(x) = \\frac{\\sin(\\pi x)}{x^2 - 2x + 2} + e^..."
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  "content": "设函数\n \nf(x) = \\frac{\\sin(\\pi x)}{x^2 - 2x + 2} + e^{-x^2}\\ln\\left(1 + \\frac{1}{1+x^2}\\right)\n \n定义\n \nI = \\lim_{n\\to\\infty} \\int_{-n}^{n} \\left( f(x) - \\frac12 f''(x) \\right) \\arctan\\left(e^x\\right) dx\n\n再令\n \nJ = \\sum_{k=1}^{\\infty} \\frac{(-1)^{k+1}}{k} \\int_{0}^{1} \\frac{x^{k-1}}{\\sqrt{1 - x^2}} dx\n \n求\n\nK = I + J\n\n并给出 K的精确值。",
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