\u4ee5\u4e0b\u306e\u5404\u554f\u306b\u305d\u308c\u305e\u308c\u7b54\u3048\u3088.<\/p>\r\n
\u554f1.<\/strong>\u3000\u5b9a\u7a4d\u5206 \\(\\displaystyle\\int _ 0^2 \\dfrac{2x+1}{\\sqrt{x^2+4}} \\, dx\\) \u3092\u6c42\u3081\u3088.<\/p><\/li>\r\n \u554f2.<\/strong>\u3000\\(1\\) \u6b69\u3067 \\(1\\) \u6bb5\u307e\u305f\u306f \\(2\\) \u6bb5\u306e\u3044\u305a\u308c\u304b\u3067\u968e\u6bb5\u3092\u6607\u308b\u3068\u304d, \\(1\\) \u6b69\u3067 \\(2\\) \u6bb5\u6607\u308b\u3053\u3068\u306f\u9023\u7d9a\u3057\u306a\u3044\u3082\u306e\u3068\u3059\u308b. \\(15\\) \u6bb5\u306e\u968e\u6bb5\u3092\u6607\u308b\u6607\u308a\u65b9\u306f\u4f55\u901a\u308a\u3042\u308b\u304b.<\/p><\/li>\r\n<\/ol>\r\n \u554f1.<\/strong><\/p>\r\n \u6c42\u3081\u308b\u5b9a\u7a4d\u5206\u3092 \\(I\\) \u3068\u304a\u304f\u3068\r\n\\[\r\nI = \\underline{\\displaystyle\\int _ 0^2 \\dfrac{( x^2+4 )'}{\\sqrt{x^2+4}} \\, dx} _ {[1]} +\\underline{\\displaystyle\\int _ 0^2 \\dfrac{1}{\\sqrt{x^2+4}} \\, dx} _ {[2]}\r\n\\]\r\n\u4e0b\u7dda\u90e8 [1] \u306b\u3064\u3044\u3066\r\n\\[\\begin{align}\r\n[1] & = \\left[ 2 \\sqrt{x^2+4} \\right] _ 0^2 \\\\\r\n& = 4 \\left( \\sqrt{2} -1 \\right)\r\n\\end{align}\\]\r\n\u4e0b\u7dda\u90e8 [2] \u306b\u3064\u3044\u3066, \\(x = 2 \\tan \\theta \\ \\left( -\\dfrac{\\pi}{2} \\lt \\theta \\lt \\dfrac{\\pi}{2} \\right)\\) \u3068\u304a\u304f\u3068\r\n\\[\\begin{gather}\r\ndx = \\dfrac{2}{\\cos^2 \\theta} d \\theta , \\\\\r\n\\begin{array}{c|ccc} x & 0 & \\rightarrow & 2 \\\\ \\hline \\theta & 0 & \\rightarrow & \\dfrac{\\pi}{4} \\end{array}\r\n\\end{gather}\\]\r\n\u306a\u306e\u3067\r\n\\[\\begin{align}\r\n[2] & = \\displaystyle\\int _ {0}^{\\frac{\\pi}{4}} \\dfrac{1}{2 \\sqrt{\\tan^2 \\theta +1}} \\cdot \\dfrac{2}{\\cos^2 \\theta} \\, d \\theta \\\\\r\n& = \\displaystyle\\int _ {0}^{\\frac{\\pi}{4}} \\dfrac{1}{\\cos \\theta} \\, d \\theta \\\\\r\n& = \\displaystyle\\int _ {0}^{\\frac{\\pi}{4}} \\dfrac{\\cos \\theta}{1 -\\sin^2 \\theta} \\, d \\theta \\\\\r\n& = \\dfrac{1}{2} \\displaystyle\\int _ {0}^{\\frac{\\pi}{4}} \\left\\{ -\\dfrac{( 1 -\\sin \\theta )'}{1 -\\sin \\theta} +\\dfrac{( 1 +\\sin \\theta )'}{1 +\\sin \\theta} \\right\\} \\, d \\theta \\\\\r\n& = \\dfrac{1}{2} \\left[ \\log \\left| \\dfrac{1 +\\sin \\theta}{1 -\\sin \\theta} \\right| \\right] _ {0}^{\\frac{\\pi}{4}} \\\\\r\n& = \\dfrac{1}{2} \\log \\dfrac{\\sqrt{2} +1}{\\sqrt{2} -1} \\\\\r\n& = \\log \\left( \\sqrt{2} +1 \\right)\r\n\\end{align}\\]\r\n\u4ee5\u4e0a\u3088\u308a\r\n\\[\r\nI = \\underline{4 \\left( \\sqrt{2} -1 \\right) +\\log \\left( \\sqrt{2} +1 \\right)}\r\n\\]\r\n \u554f2.<\/strong><\/p>\r\n \\(n\\) \u6bb5\u76ee\u306b\u9054\u3059\u308b\u65b9\u6cd5\u306e\u6570\u3092 \\(c _ n\\) \u3068\u304a\u304d, \u305d\u306e\u3046\u3061, \\(1\\) \u6bb5\u6607\u3063\u3066\u9054\u3059\u308b\u65b9\u6cd5\u306e\u6570\u3092 \\(a _ n\\) , \\(2\\) \u6bb5\u6607\u3063\u3066\u9054\u3059\u308b\u65b9\u6cd5\u306e\u6570\u3092 \\(b _ n\\) \u3068\u3059\u308b.\r\n\\[\r\nc _ n = a _ n +b _ n \\quad ... [1]\r\n\\]\r\n\\(2\\) \u6bb5\u9023\u7d9a\u3067\u6607\u308b\u3053\u3068\u306f\u3067\u304d\u306a\u3044\u306e\u3067, \\(n \\geqq 1\\) \u306b\u3064\u3044\u3066\r\n\\[\r\na _ {n+1} = a _ n +b _ n , \\ b _ {n+2} = a _ n\r\n\\]\r\n\u3057\u305f\u304c\u3063\u3066\r\n\\[\r\na _ {n+3} = a _ {n+2} +a _ n \\quad ... [2]\r\n\\]\r\n\u307e\u305f, [1] \u3088\u308a\r\n\\[\r\nc _ {n+3} = a _ {n+2} +a _ {n+1} +a _ n \\quad ... [3]\r\n\\]\r\n\\(a _ 1 = 1\\) , \\(a _ 2 = 1\\) , \\(a _ 3 = 2\\) \u3068[2]\u3088\u308a\r\n\\[\r\n\\begin{array}{c|cccccccccccccc} n & 1 & 2 & 3 & 4 & 5 & 6 & 7 & 8 & 9 & 10 & 11 & 12 & 13 & 14 \\\\ \\hline a _ n & 1 & 1 & 2 & 3 & 4 & 6 & 9 & 13 & 19 & 28 & 41 & 60 & 88 & 129 \\end{array}\r\n\\]\r\n\u3088\u3063\u3066, [3] \u3088\u308a\r\n\\[\\begin{align}\r\nc _ {15} & = a _ {14} +a _ {13} +a _ {12} \\\\\r\n& = 129 +88 +60 =\\underline{277}\r\n\\end{align}\\]\r\n","protected":false},"excerpt":{"rendered":"\u4ee5\u4e0b\u306e\u5404\u554f\u306b\u305d\u308c\u305e\u308c\u7b54\u3048\u3088. \u554f1.\u3000\u5b9a\u7a4d\u5206 \\(\\displaystyle\\int _ 0^2 \\dfrac{2x+1}{\\sqrt{x^2+4}} \\, dx\\) \u3092\u6c42\u3081\u3088. \u554f2.\u3000\\(1\\) \u6b69\u3067 \\(1\\) \u6bb5 … \u7d9a\u304d\u3092\u8aad\u3080
\r\n\r\n\u3010 \u89e3 \u7b54 \u3011<\/h2>\r\n