\u5b9f\u6570 \\(x\\) \u306b\u5bfe\u3057\u3066, \\(f(x) = \\displaystyle\\int _ 0^{\\frac{\\pi}{2}} \\left| \\cos t -x \\sin 2t \\right| \\, dt\\) \u3068\u304a\u304f.<\/p>\r\n
(1)<\/strong>\u3000\u95a2\u6570 \\(f(x)\\) \u306e\u6700\u5c0f\u5024\u3092\u6c42\u3081\u3088.<\/p><\/li>\r\n (2)<\/strong>\u3000\u5b9a\u7a4d\u5206 \\(\\displaystyle\\int _ 0^1 f(x) \\, dx\\) \u3092\u6c42\u3081\u3088.<\/p><\/li>\r\n<\/ol>\r\n (1)<\/strong><\/p>\r\n \\(g(x) = \\cos t -x \\sin 2t\\) , \\(G(x) = \\displaystyle\\int g(x) \\, dx\\) \u3068\u304a\u304f.\r\n\\[\\begin{align}\r\ng(x) & = \\cos t -2x \\sin t \\cos t = \\cos t \\left( 1 -2x \\sin t \\right) \\\\\r\nG(x) & = \\sin t +\\dfrac{x}{2} \\cos 2t +C \\quad ( C \\text{\u306f\u7a4d\u5206\u5b9a\u6570} )\r\n\\end{align}\\]\r\n 1*<\/strong>\u3000\\(2x \\leqq 1\\) \u3059\u306a\u308f\u3061 \\(x \\leqq \\dfrac{1}{2}\\) \u306e\u3068\u304d 2*<\/strong>\u3000\\(2x \\gt 1\\) \u3059\u306a\u308f\u3061 \\(x \\gt \\dfrac{1}{2}\\) \u306e\u3068\u304d 1*<\/strong> 2*<\/strong>\u3088\u308a, \u6c42\u3081\u308b\u6700\u5c0f\u5024\u306f\r\n\\[\r\n\\underline{\\sqrt{2}-1}\r\n\\]\r\n (2)<\/strong><\/p>\r\n (1)<\/strong> \u306e\u904e\u7a0b\u3088\u308a\r\n\\[\\begin{align}\r\n\\displaystyle\\int _ 0^1 f(x) \\, dx & = \\displaystyle\\int _ 0^{\\frac{1}{2}} (1-x) \\, dx +\\displaystyle\\int _ {\\frac{1}{2}}^1 \\left( x +\\dfrac{1}{2x} -1 \\right) \\, dx \\\\\r\n& = \\left[ x -\\dfrac{x^2}{2} \\right] _ 0^{\\frac{1}{2}} +\\left[ \\dfrac{x^2}{2} +\\dfrac{\\log x}{2} -x \\right] _ {\\frac{1}{2}}^1 \\\\\r\n& = \\left( \\dfrac{1}{2} -\\dfrac{1}{8} \\right) -\\dfrac{1}{2} -\\left( \\dfrac{1}{8} -\\dfrac{\\log 2}{2} -\\dfrac{1}{2} \\right) \\\\\r\n& = \\underline{\\dfrac{1}{4} +\\dfrac{\\log 2}{2}}\r\n\\end{align}\\]\r\n","protected":false},"excerpt":{"rendered":"\u5b9f\u6570 \\(x\\) \u306b\u5bfe\u3057\u3066, \\(f(x) = \\displaystyle\\int _ 0^{\\frac{\\pi}{2}} \\left| \\cos t -x \\sin 2t \\right| \\, dt\\) \u3068\u304a\u304f. (1 … \u7d9a\u304d\u3092\u8aad\u3080
\r\n\r\n\u3010 \u89e3 \u7b54 \u3011<\/h4>\r\n
\r\n
\r\n\\(0 \\leqq t \\leqq \\dfrac{\\pi}{2}\\) \u306b\u304a\u3044\u3066, \\(g(t) \\geqq 0\\) \u306a\u306e\u3067\r\n\\[\\begin{align}\r\nf(x) & = G\\left( \\dfrac{\\pi}{2} \\right) -G(0) \\\\\r\n& = \\left( 1-\\dfrac{x}{2} \\right) -\\dfrac{x}{2} \\\\\r\n& = 1-x \\geqq \\dfrac{1}{2}\r\n\\end{align}\\]<\/li>\r\n
\r\n\\(\\sin \\alpha = \\dfrac{1}{2x} \\ \\left( 0 \\leqq \\alpha \\leqq \\dfrac{\\pi}{2} \\right) \\quad ... [1]\\) \u3092\u307f\u305f\u3059. \\(\\alpha\\) \u3092\u304a\u304f\u3068, \\(g(t)\\) \u306e\u7b26\u53f7\u306f\u4e0b\u8868\u306e\u3088\u3046\u306b\u306a\u308b.\r\n\\[\r\n\\begin{array}{c|ccccc} t & 0 & \\cdots & \\alpha & \\cdots & \\dfrac{\\pi}{2} \\\\ \\hline g(t) & & + & 0 & - & \\end{array}\r\n\\]\r\n\u3057\u305f\u304c\u3063\u3066\r\n\\[\\begin{align}\r\nf(x) & = \\left\\{ G( \\alpha ) -G(0) \\right\\} -\\left\\{ G \\left( \\dfrac{\\pi}{2} \\right) -G( \\alpha ) \\right\\} \\\\\r\n& = -\\dfrac{x}{2} -\\left( 1-\\dfrac{x}{2} \\right) +2 \\left( \\sin \\alpha +\\dfrac{x}{2} \\cos 2\\alpha \\right) \\\\\r\n& = -1 +2 \\left\\{ \\dfrac{1}{2x} +\\dfrac{x}{2} \\left( 1 -2 \\cdot \\dfrac{1}{4x^2} \\right) \\right\\} \\quad ( \\ \\text{\u2235} \\ [1] \\ ) \\\\\r\n& = x +\\dfrac{1}{2x} -1 \\geqq 2\\sqrt{x \\cdot \\dfrac{1}{2x}} -1 \\quad ( \\ \\text{\u2235} \\ \\text{ \u76f8\u52a0\u76f8\u4e57\u5e73\u5747\u306e\u95a2\u4fc2} ) \\\\\r\n& = \\sqrt{2}-1 \\lt \\dfrac{1}{2}\r\n\\end{align}\\]\r\n\u7b49\u53f7\u6210\u7acb\u306f, \\(x = \\dfrac{1}{2x}\\) \u3059\u306a\u308f\u3061 \\(x = \\dfrac{\\sqrt{2}}{2}\\) \u306e\u3068\u304d<\/p><\/li>\r\n<\/ol>\r\n