(1)<\/strong>\u3000\\(\\displaystyle\\int e^{-x} \\cos x \\, dx = e^{-x} \\left( p \\sin x +q \\cos x \\right) +C\\) \u3092\u307f\u305f\u3059\u5b9a\u6570 \\(p , q\\) \u3092\u6c42\u3081\u3088. \u305f\u3060\u3057, \\(C\\) \u306f\u7a4d\u5206\u5b9a\u6570\u3067\u3042\u308b.<\/p><\/li>\r\n (2)<\/strong>\u3000\\(a _ 1\\) \u306e\u5024\u3092\u6c42\u3081\u3088.<\/p><\/li>\r\n (3)<\/strong>\u3000\\(a _ n\\) \u306e\u5024\u3092\u6c42\u3081\u3088.<\/p><\/li>\r\n (4)<\/strong>\u3000\\(\\displaystyle\\lim _ {n \\rightarrow \\infty} \\left( a _ 1 +a _ 2 + \\cdots + a _ n \\right)\\) \u3092\u6c42\u3081\u3088.<\/p><\/li>\r\n<\/ol>\r\n (1)<\/strong><\/p>\r\n \u90e8\u5206\u7a4d\u5206\u3092\u7e70\u308a\u8fd4\u3057\u7528\u3044\u308c\u3070\r\n\\[\\begin{align}\r\n\\displaystyle\\int e^{-x} \\cos x \\, dx & = e^{-x} \\sin x +\\displaystyle\\int e^{-x} \\sin x \\, dx \\\\\r\n& = e^{-x} \\sin x -e^{-x} \\cos x -\\displaystyle\\int e^{-x} \\cos x \\, dx \\\\\r\n\\text{\u2234} \\quad \\displaystyle\\int e^{-x} \\cos x \\, dx & = \\dfrac{e^{-x}}{2} \\left( \\sin x -\\cos x \\right) +C \\quad ( \\ C \\text{\u306f\u7a4d\u5206\u5b9a\u6570} )\r\n\\end{align}\\]\r\n\u3088\u3063\u3066\r\n\\[\r\np = \\underline{\\dfrac{1}{2}} , \\ q = \\underline{-\\dfrac{1}{2}}\r\n\\]\r\n (2)<\/strong><\/p>\r\n \\(F(x) = \\dfrac{e^{-x}}{2} \\left( \\sin x -\\cos x \\right)\\) \u3068\u304a\u3044\u3066, (1)<\/strong> \u306e\u7d50\u679c\u3092\u7528\u3044\u308c\u3070\r\n\\[\\begin{align}\r\na _ 1 & = \\displaystyle\\int _ 0^{\\pi} e^{-x} \\left( 1 -\\left| \\cos x \\right| \\right) \\, dx \\\\\r\n& = \\displaystyle\\int _ 0^{\\pi} e^{-x} \\, dx -\\displaystyle\\int _ 0^{\\frac{\\pi}{2}} e^{-x} \\cos x \\, dx +\\displaystyle\\int _ {\\frac{\\pi}{2}}^{\\pi} e^{-x} \\cos x \\, dx \\\\\r\n& = \\left[ -e^{-x} \\right] _ 0^{\\pi} +F(0) +F( \\pi ) -2 F \\left( \\dfrac{\\pi}{2} \\right) \\\\\r\n& = 1 -e^{-\\pi} -\\dfrac{1}{2} +\\dfrac{e^{-\\pi}}{2} -e^{-\\frac{\\pi}{2}} \\\\\r\n& = \\underline{\\dfrac{e^{\\pi} -2 e^{\\frac{\\pi}{2}} -1}{2 e^{\\pi}}}\r\n\\end{align}\\]\r\n (3)<\/strong><\/p>\r\n \\(y = \\left| \\cos x \\right|\\) \u306f\u5468\u671f \\(\\pi\\) \u306e\u5468\u671f\u95a2\u6570\u306a\u306e\u3067\r\n\\[\r\na _ {n+1} = \\displaystyle\\int _ {n \\pi}^{(n+1) \\pi} e^{-x} \\left( 1 -\\left| \\cos x \\right| \\right) \\, dx\r\n\\]\r\n\u306b\u5bfe\u3057\u3066, \\(u = x -\\pi\\) \u3068\u304a\u304f\u3068, \\(du = dx\\) \u3067\r\n\\[\r\n\\begin{array}{c|ccc} x & n \\pi & \\rightarrow & (n+1) \\pi \\\\ \\hline u & (n-1) \\pi & \\rightarrow & n \\pi \\end{array}\r\n\\]\r\n\u306a\u306e\u3067\r\n\\[\\begin{align}\r\na _ {n+!} & = \\displaystyle\\int _ {(n-1) \\pi}^{n \\pi} e^{-( u +\\pi )} \\left\\{ 1 -\\left| \\cos ( u +\\pi ) \\right| \\right\\} \\, du \\\\\r\n& = e^{-\\pi} \\displaystyle\\int _ {(n-1) \\pi}^{n \\pi} e^{-u} \\left( 1 -\\left| \\cos u \\right| \\right) \\, du \\\\\r\n& = e^{-\\pi} a _ n\r\n\\end{align}\\]\r\n\u3088\u3063\u3066, \u6570\u5217 \\(\\{ a _ n \\}\\) \u306f\u521d\u9805 \\(a _ 1 = \\dfrac{e^{\\pi} -2 e^{\\frac{\\pi}{2}} -1}{2 e^{\\pi}}\\) , \u516c\u6bd4 \\(e^{-\\pi}\\) \u306e\u7b49\u6bd4\u6570\u5217\u306a\u306e\u3067\r\n\\[\\begin{align}\r\na _ n & = \\dfrac{e^{\\pi} -2 e^{\\frac{\\pi}{2}} -1}{2 e^{\\pi}} \\cdot e^{-(n-1) \\pi} \\\\\r\n& = \\underline{\\dfrac{e^{\\pi} -2 e^{\\frac{\\pi}{2}} -1}{2 e^{n \\pi}}}\r\n\\end{align}\\]\r\n (4)<\/strong><\/p>\r\n (3)<\/strong> \u306e\u7d50\u679c\u3088\u308a\r\n\\[\\begin{align}\r\n\\displaystyle\\lim _ {n \\rightarrow \\infty} \\left( a _ 1 +a _ 2 + \\cdots + a _ n \\right) & = \\dfrac{e^{\\pi} -2 e^{\\frac{\\pi}{2}} -1}{2 e^{\\pi}} \\cdot \\dfrac{1}{1 -e^{-\\pi}} \\\\\r\n& = \\underline{\\dfrac{e^{\\pi} -2 e^{\\frac{\\pi}{2}} -1}{2 \\left( e^{\\pi} -1 \\right)}}\r\n\\end{align}\\]\r\n","protected":false},"excerpt":{"rendered":"\u66f2\u7dda \\(y = e^{-x}\\) \u3068 \\(y = e^{-x} \\left| \\cos x \\right|\\) \u3067\u56f2\u307e\u308c\u305f\u56f3\u5f62\u306e\u3046\u3061, \\((n-1) \\pi \\leqq x \\leqq n \\pi\\) \u3092\u307f\u305f\u3059\u90e8\u5206\u306e … \u7d9a\u304d\u3092\u8aad\u3080
\r\n\r\n\u3010 \u89e3 \u7b54 \u3011<\/h2>\r\n
![\"waseda_r_2007_03_01\"](\"\/\/www.roundown.net\/nyushi\/wp-content\/uploads\/waseda_r_2007_03_01.png\")
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