\u6b21\u306e\u554f\u3044\u306b\u7b54\u3048\u3088.<\/p>\r\n
(1)<\/strong>\u3000\u5b9a\u7a4d\u5206 \\(\\displaystyle\\int _ 0^{\\sqrt{\\frac{\\pi}{2}}} x^3 \\cos \\left( x^2 \\right) \\, dx\\) \u3092\u6c42\u3081\u3088.<\/p><\/li>\r\n (2)<\/strong>\u3000\\(0 \\lt x \\lt 1\\) \u306e\u3068\u304d, \u4e0d\u7b49\u5f0f\r\n\\[\r\n\\left( \\dfrac{x+1}{2} \\right)^{x+1} \\lt x^x\r\n\\]\r\n\u304c\u6210\u308a\u7acb\u3064\u3053\u3068\u3092\u793a\u305b.<\/p><\/li>\r\n<\/ol>\r\n (1)<\/strong><\/p>\r\n \\(t = x^2\\) \u3068\u304a\u304f\u3068\r\n\\[\r\ndt = 2x \\, dx , \\quad \\begin{array}{c|ccc} x & 0 & \\rightarrow & \\sqrt{\\dfrac{\\pi}{2}} \\\\ \\hline t & 0 & \\rightarrow & \\dfrac{\\pi}{2} \\end{array}\r\n\\]\r\n\u3088\u3063\u3066, \u6c42\u3081\u308b\u7a4d\u5206 \\(I\\) \u306f\r\n\\[\\begin{align}\r\nI & = \\dfrac{1}{2} \\displaystyle\\int _ 0^{\\sqrt{\\frac{\\pi}{2}}} x^2 \\cos x^2 \\cdot 2x \\, dx \\\\\r\n& = \\dfrac{1}{2} \\displaystyle\\int _ 0^{\\frac{\\pi}{2}} t \\cos t \\, dt \\\\\r\n& = \\dfrac{1}{2} \\left[ t \\sin t \\right] _ 0^{\\frac{\\pi}{2}} -\\dfrac{1}{2} \\displaystyle\\int _ 0^{\\frac{\\pi}{2}} \\sin t \\, dt \\\\\r\n& = \\dfrac{\\pi}{4} +\\dfrac{1}{2} \\left[ \\cos t \\right] _ 0^{\\frac{\\pi}{2}} \\\\\r\n& = \\underline{\\dfrac{\\pi}{4} -\\dfrac{1}{2}}\r\n\\end{align}\\]\r\n (2)<\/strong><\/p>\r\n \u793a\u3059\u3079\u304d\u4e0d\u7b49\u5f0f\u306b\u3064\u3044\u3066, \u4e21\u8fba\u5bfe\u6570\u3092\u3068\u3063\u3066\u5909\u5f62\u3059\u308b\u3068\r\n\\[\\begin{align}\r\n(x+1) \\left\\{ \\log (x+1) -\\log 2 \\right\\} & \\lt x \\log x \\\\\r\n(x+1) \\left\\{ \\log (x+1) -\\log 2 \\right\\} -x \\log x & \\lt 0 \\quad ... [ \\text{A} ]\r\n\\end{align}\\]\r\n\u3057\u305f\u304c\u3063\u3066, [A] \u304c\u6210\u7acb\u3059\u308b\u3053\u3068\u3092\u793a\u305b\u3070\u3088\u3044.
\r\n\r\n\u3010 \u89e3 \u7b54 \u3011<\/h4>\r\n
\r\n[A] \u306e\u5de6\u8fba\u3092 \\(f(x)\\) \u3068\u304a\u304f\u3068\r\n\\[\\begin{align}\r\nf'(x) & = \\left\\{ \\log (x+1) -\\log 2 \\right\\} +1 -\\log x -1 \\\\\r\n& = \\log \\dfrac{x+1}{2x} \\\\\r\n& = \\log \\dfrac{1}{2} \\left( 1 +\\dfrac{1}{x} \\right) \\\\\r\n& \\gt \\log 1 = 0 \\quad \\left( \\ \\text{\u2235} \\ \\dfrac{1}{x} \\gt 1 \\ \\right)\r\n\\end{align}\\]\r\n\u3086\u3048\u306b, \\(0 \\lt x \\lt 1\\) \u306b\u304a\u3044\u3066, \\(f(x)\\) \u306f\u5358\u8abf\u5897\u52a0\u3057\r\n\\[\r\nf(x) \\lt f(1) = 0\r\n\\]\r\n\u3088\u3063\u3066, [A] \u304c\u793a\u3055\u308c, \u984c\u610f\u3082\u793a\u3055\u308c\u305f.<\/p>\r\n","protected":false},"excerpt":{"rendered":"\u6b21\u306e\u554f\u3044\u306b\u7b54\u3048\u3088. (1)\u3000\u5b9a\u7a4d\u5206 \\(\\displaystyle\\int _ 0^{\\sqrt{\\frac{\\pi}{2}}} x^3 \\cos \\left( x^2 \\right) \\, dx\\) \u3092\u6c42\u3081\u3088. (2) … \u7d9a\u304d\u3092\u8aad\u3080