\u6b21\u306e\u554f\u3044\u306b\u7b54\u3048\u3088.<\/p>\r\n
(1)<\/strong>\u3000\u4e0d\u5b9a\u7a4d\u5206\r\n\\[\r\n\\displaystyle\\int x^2 \\cos \\left( a \\log x \\right) \\, dx\r\n\\]\r\n\u3092\u6c42\u3081\u3088. \u305f\u3060\u3057, \\(a\\) \u306f \\(0\\) \u3067\u306a\u3044\u5b9a\u6570\u3068\u3059\u308b.<\/p><\/li>\r\n (2)<\/strong>\u3000\u66f2\u7dda \\(y = x \\cos \\left( \\log x \\right)\\) \u3068 \\(x\\) \u8ef8, \u304a\u3088\u3073 \\(2\\) \u76f4\u7dda \\(x = 1\\) , \\(x = e^{\\frac{\\pi}{4}}\\) \u3067\u56f2\u307e\u308c\u305f\u56f3\u5f62\u3092, \\(x\\) \u8ef8\u306e\u307e\u308f\u308a\u306b \\(1\\) \u56de\u8ee2\u3057\u3066\u3067\u304d\u308b\u56de\u8ee2\u4f53\u306e\u4f53\u7a4d\u3092\u6c42\u3081\u3088.<\/p><\/li>\r\n<\/ol>\r\n (1)<\/strong><\/p>\r\n \u4ee5\u4e0b\u3067\u306f, \\(C\\) \u306f\u7a4d\u5206\u5b9a\u6570\u3092\u8868\u3059. (2)<\/strong><\/p>\r\n \u6c42\u3081\u308b\u4f53\u7a4d \\(V\\) \u306f, (1)<\/strong> \u306e\u7d50\u679c\u3092\u7528\u3044\u3066\r\n\\[\\begin{align}\r\nV & = \\pi \\displaystyle\\int _ {1}^{e^{\\frac{\\pi}{4}}} x^2 \\cos^2 \\left( \\log x \\right) \\, dx \\\\\r\n& =\\pi \\displaystyle\\int _ {1}^{e^{\\frac{\\pi}{4}}} x^2 \\cdot \\dfrac{1 +\\cos \\left( 2 \\log x \\right)}{2} \\, dx \\\\\r\n& =\\dfrac{\\pi}{2} \\left[ \\dfrac{x^3 \\left\\{ 2 \\sin \\left( 2 \\log x \\right) +3 \\cos \\left( 2 \\log x \\right) \\right\\}}{2^2+9} \\right] _ {1}^{e^{\\frac{\\pi}{4}}} \\\\\r\n& \\qquad +\\dfrac{\\pi}{2} \\left[ \\dfrac{x^3}{3} \\right] _ {1}^{e^{\\frac{\\pi}{4}}} \\\\\r\n& = \\dfrac{\\pi}{2} \\left( \\dfrac{2 e^{\\frac{3 \\pi}{4}}}{13} -\\dfrac{3}{13} \\right) +\\dfrac{\\pi}{6} \\left( e^{\\frac{3 \\pi}{4}} -1 \\right) \\\\\r\n& = \\underline{\\dfrac{\\pi \\left( 19 e^{\\frac{3 \\pi}{4}} +22 \\right)}{78}}\r\n\\end{align}\\]\r\n","protected":false},"excerpt":{"rendered":"\u6b21\u306e\u554f\u3044\u306b\u7b54\u3048\u3088. (1)\u3000\u4e0d\u5b9a\u7a4d\u5206 \\[ \\displaystyle\\int x^2 \\cos \\left( a \\log x \\right) \\, dx \\] \u3092\u6c42\u3081\u3088. \u305f\u3060\u3057, \\(a\\) \u306f \\(0\\) \u3067\u306a\u3044 … \u7d9a\u304d\u3092\u8aad\u3080
\r\n\r\n\u3010 \u89e3 \u7b54 \u3011<\/h4>\r\n
\r\n\\(I _ c = \\displaystyle\\int x^2 \\cos \\left( a \\log x \\right) \\, dx\\) , \\(I _ s = \\displaystyle\\int x^2 \\sin \\left( a \\log x \\right) \\, dx\\) \u3068\u304a\u304f.
\r\n\\((a \\log x)' =\\dfrac{a}{x}\\) \u306a\u306e\u3067\r\n\\[\\begin{align}\r\n\\displaystyle\\int \\dfrac{a}{x} \\cos \\left( a \\log x \\right) \\, dx & = \\sin \\left( a \\log x \\right) +C , \\\\\r\n\\displaystyle\\int \\dfrac{a}{x} \\sin \\left( a \\log x \\right) \\, dx & = -\\cos \\left( a \\log x \\right) +C\r\n\\end{align}\\]\r\n\u3053\u308c\u3092\u7528\u3044\u3066, \\(I _ c , I _ s\\) \u3092\u90e8\u5206\u7a4d\u5206\u3059\u308b\u3068\r\n\\[\\begin{align}\r\nI _ c & = \\dfrac{x^3}{a} \\sin \\left( a \\log x \\right) -\\displaystyle\\int \\dfrac{3x^2}{a} \\sin \\left( a \\log x \\right) \\, dx \\\\\r\n& = \\dfrac{x^3}{a} \\sin \\left( a \\log x \\right) -\\dfrac{3}{a} I _ s \\\\\r\n\\text{\u2234} \\quad a I _ c & +3 I _ s = x^3 \\sin \\left( a \\log x \\right) \\quad ... [1] , \\\\\r\nI _ s & = -\\dfrac{x^3}{a} \\cos \\left( a \\log x \\right) +\\displaystyle\\int \\dfrac{3x^2}{a} \\sin \\left( a \\log x \\right) \\, dx \\\\\r\n& = -\\dfrac{x^3}{a} \\cos \\left( a \\log x \\right) +\\dfrac{3}{a} I _ c \\\\\r\n\\text{\u2234} \\quad 3 I _ c & -a I _ s = x^3 \\cos \\left( a \\log x \\right) \\quad ... [2]\r\n\\end{align}\\]\r\n[1] [2] \u3088\u308a\r\n\\[\r\nI _ c =\\underline{\\dfrac{x^3 \\left\\{ a \\sin \\left( a \\log x \\right) +3 \\cos \\left( a \\log x \\right) \\right\\}}{a^2+9} +C}\r\n\\]\r\n