{"id":473,"date":"2012-11-18T18:28:29","date_gmt":"2012-11-18T09:28:29","guid":{"rendered":"http:\/\/www.roundown.net\/nyushi\/?p=473"},"modified":"2021-09-25T19:42:23","modified_gmt":"2021-09-25T10:42:23","slug":"thr201204","status":"publish","type":"post","link":"https:\/\/www.roundown.net\/nyushi\/thr201204\/","title":{"rendered":"\u6771\u5317\u5927\u7406\u7cfb2012\uff1a\u7b2c4\u554f"},"content":{"rendered":"<hr \/>\n<p>\\(0 \\leqq x \\leqq \\pi\\) \u306b\u5bfe\u3057\u3066, \u95a2\u6570 \\(f(x)\\) \u3092\r\n\\[\r\nf(x) = \\displaystyle\\int _ {0}^{\\frac{\\pi}{2}} \\dfrac{\\cos |t-x|}{1 +\\sin |t-x|} \\, dt\r\n\\]\r\n\u3068\u5b9a\u3081\u308b.\r\n\\(f(x)\\) \u306e \\(0 \\leqq x \\leqq \\pi\\) \u306b\u304a\u3051\u308b\u6700\u5927\u5024\u3068\u6700\u5c0f\u5024\u3092\u6c42\u3081\u3088.<\/p>\r\n<hr \/>\r\n<!--more-->\r\n<h4>\u3010 \u89e3 \u7b54 \u3011<\/h4>\r\n<p>\\[\\begin{align}\r\n\\cos |t-x| & = \\cos (t-x) \\\\\r\n\\sin |t-x| & = \\left\\{ \\begin{array}{ll} \\sin (t-x) & \\left( \\ t \\geqq x \\text{\u306e\u3068\u304d} \\right) \\\\ -\\sin (t-x) & \\left( \\ t \\lt x \\text{\u306e\u3068\u304d} \\right) \\end{array} \\right.\r\n\\end{align}\\]\r\n\u307e\u305f\r\n\\[\\begin{align}\r\n\\displaystyle\\int \\dfrac{\\cos (t-x)}{1 \\pm \\sin (t-x)} \\, dt & = \\displaystyle\\int \\dfrac{\\pm \\left\\{ 1 \\pm \\sin (t-x) \\right\\}'}{1 \\pm \\sin (t-x)} \\, dt \\\\\r\n& = \\pm \\log \\left\\{ 1 \\pm \\sin (t-x) \\right\\} +C \\quad ( \\ C \\text{\u306f\u7a4d\u5206\u5b9a\u6570} )\r\n\\end{align}\\]\r\n\u4ee5\u4e0a\u3092\u7528\u3044\u3066, \u5834\u5408\u5206\u3051\u3057\u3066\u8003\u3048\u308b.<\/p>\r\n<ol>\r\n<li><p><strong>1*<\/strong>\u3000\\(0 \\leqq x \\leqq \\dfrac{\\pi}{2}\\) \u306e\u3068\u304d\r\n\\[\\begin{align}\r\nf(x) & =\\displaystyle\\int _ {0}^{x} \\dfrac{\\cos (t-x)}{1 -\\sin (t-x)} \\, dt +\\displaystyle\\int _ {x}^{\\frac{\\pi}{2}} \\dfrac{\\cos (t-x)}{1 +\\sin (t-x)} \\, dt \\\\\r\n& = \\left[ -\\log \\left\\{ 1 - \\sin (t-x) \\right\\} \\right] _ {0}^{x} +\\left[ \\log \\left\\{ 1 + \\sin (t-x) \\right\\} \\right] _ {x}^{\\frac{\\pi}{2}} \\\\\r\n& = \\log ( 1+\\sin x ) +\\log \\left\\{ 1 +\\sin \\left( \\dfrac{\\pi}{2} -x \\right) \\right\\} \\\\\r\n& = \\log \\underline{( 1+\\sin x )( 1+\\cos x )} _ {[1]}\r\n\\end{align}\\]\r\n\u95a2\u6570 \\(\\log x\\) \u306f\u5358\u8abf\u5897\u52a0\u306a\u306e\u3067, \u4e0b\u7dda\u90e8 [1] \u3092 \\(g(x)\\) \u3068\u304a\u3044\u3066, \u3053\u308c\u306e\u5897\u6e1b\u3092\u8abf\u3079\u308c\u3070\u3088\u3044.\r\n\\[\\begin{align}\r\ng'(x) & = \\cos x ( 1+\\cos x ) -\\sin x ( 1+\\sin x ) \\\\\r\n& = ( \\cos x -\\sin x )( \\cos x +\\sin x +1 ) \\\\\r\n& = \\cos \\left( x +\\dfrac{\\pi}{4} \\right) \\left\\{ \\sin \\left( x +\\dfrac{\\pi}{4} \\right) +1 \\right\\}\r\n\\end{align}\\]\r\n\\(g'(x)=0\\) \u3092\u89e3\u304f\u3068\r\n\\[\r\nx = \\dfrac{\\pi}{4}\r\n\\]\r\n\\(0 \\leqq x \\leqq \\dfrac{\\pi}{2}\\) \u306b\u304a\u3051\u308b \\(g(x)\\) \u306e\u5897\u6e1b\u306f\u4e0b\u8868\u306e\u3068\u304a\u308a.\r\n\\[\r\n\\begin{array}{c|ccccc} x & 0 & \\cdots & \\frac{\\pi}{4} & \\cdots & \\frac{\\pi}{2}\\\\ \\hline g'(x) & & + & 0 & - & \\\\ \\hline g(x) & 2 & \\nearrow & \\left( 1 +\\frac{\\sqrt{2}}{2} \\right)^2 & \\searrow & 2 \\end{array}\r\n\\]\r\n\u3053\u3053\u3067\r\n\\[\\begin{align}\r\nf \\left( \\dfrac{\\pi}{4} \\right) & = \\log \\left( 1 +\\frac{\\sqrt{2}}{2} \\right)^2 \\\\\r\n& = 2 \\log \\dfrac{2 +\\sqrt{2}}{2}\r\n\\end{align}\\]<\/li>\r\n<li><p><strong>2*<\/strong>\u3000\\(\\dfrac{\\pi}{2} \\lt x \\leqq \\pi\\) \u306e\u3068\u304d\r\n\\[\\begin{align}\r\nf(x) & = \\displaystyle\\int _ {0}^{\\frac{\\pi}{2}} \\dfrac{\\cos (t-x)}{1 -\\sin (t-x)} \\, dt \\\\\r\n& = \\left[ -\\log \\left\\{ 1 - \\sin (t-x) \\right\\} \\right] _ {0}^{\\frac{\\pi}{2}} \\\\\r\n& = \\log ( 1+\\sin x ) -\\log \\left\\{ 1 -\\sin \\left( \\dfrac{\\pi}{2} -x \\right) \\right\\} \\\\\r\n& = \\log \\dfrac{1 +\\sin x}{1 -\\cos x}\r\n\\end{align}\\]\r\n\\(\\dfrac{\\pi}{2} \\lt x \\leqq \\pi\\) \u306b\u304a\u3044\u3066, \u95a2\u6570 \\(\\log x\\) \u306f\u5358\u8abf\u5897\u52a0, \u95a2\u6570 \\(1 +\\sin x\\) \u306f\u5358\u8abf\u6e1b\u5c11, \u95a2\u6570 \\(1 -\\cos x\\) \u306f\u5358\u8abf\u5897\u52a0\u306a\u306e\u3067, \\(f(x)\\) \u306f\u5358\u8abf\u6e1b\u5c11\u95a2\u6570\u3067\u3042\u308b.<br \/>\r\n\u3053\u3053\u3067\r\n\\[\r\nf( \\pi ) = \\log \\dfrac{1}{2} = -\\log 2\r\n\\]<\/li>\r\n<\/ol>\r\n<p><strong>1*<\/strong> <strong>2*<\/strong> \u3088\u308a, \u6c42\u3081\u308b\u6700\u5927\u5024\u3068\u6700\u5c0f\u5024\u306f\r\n\\[\r\n\\underline{\\left\\{ \\begin{array}{ll} \\text{\u6700\u5927\u5024} : \\quad & f \\left( \\dfrac{\\pi}{4} \\right) = 2 \\log \\dfrac{2 +\\sqrt{2}}{2} \\\\ \\text{\u6700\u5c0f\u5024} : \\quad & f( \\pi ) = -\\log 2 \\end{array} \\right.}\r\n\\]\r\n","protected":false},"excerpt":{"rendered":"\\(0 \\leqq x \\leqq \\pi\\) \u306b\u5bfe\u3057\u3066, \u95a2\u6570 \\(f(x)\\) \u3092 \\[ f(x) = \\displaystyle\\int _ {0}^{\\frac{\\pi}{2}} \\dfrac{\\cos |t-x &hellip; <a href=\"https:\/\/www.roundown.net\/nyushi\/thr201204\/\">\u7d9a\u304d\u3092\u8aad\u3080 <span class=\"meta-nav\">&rarr;<\/span><\/a>","protected":false},"author":1,"featured_media":0,"comment_status":"open","ping_status":"open","sticky":false,"template":"","format":"standard","meta":{"inline_featured_image":false,"footnotes":""},"categories":[72],"tags":[68,148],"class_list":["post-473","post","type-post","status-publish","format-standard","hentry","category-tohoku_r_2012","tag-68","tag-tohoku_r"],"_links":{"self":[{"href":"https:\/\/www.roundown.net\/nyushi\/wp-json\/wp\/v2\/posts\/473","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/www.roundown.net\/nyushi\/wp-json\/wp\/v2\/posts"}],"about":[{"href":"https:\/\/www.roundown.net\/nyushi\/wp-json\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"https:\/\/www.roundown.net\/nyushi\/wp-json\/wp\/v2\/users\/1"}],"replies":[{"embeddable":true,"href":"https:\/\/www.roundown.net\/nyushi\/wp-json\/wp\/v2\/comments?post=473"}],"version-history":[{"count":0,"href":"https:\/\/www.roundown.net\/nyushi\/wp-json\/wp\/v2\/posts\/473\/revisions"}],"wp:attachment":[{"href":"https:\/\/www.roundown.net\/nyushi\/wp-json\/wp\/v2\/media?parent=473"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/www.roundown.net\/nyushi\/wp-json\/wp\/v2\/categories?post=473"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/www.roundown.net\/nyushi\/wp-json\/wp\/v2\/tags?post=473"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}