{"id":423,"date":"2012-06-14T23:40:24","date_gmt":"2012-06-14T14:40:24","guid":{"rendered":"http:\/\/roundown.main.jp\/nyushi\/?p=423"},"modified":"2021-10-20T14:53:56","modified_gmt":"2021-10-20T05:53:56","slug":"ykr201201","status":"publish","type":"post","link":"https:\/\/www.roundown.net\/nyushi\/ykr201201\/","title":{"rendered":"\u6a2a\u56fd\u5927\u7406\u7cfb2012\uff1a\u7b2c1\u554f"},"content":{"rendered":"<ol>\r\n<li><p><strong>(1)<\/strong>\u3000\u5b9a\u7a4d\u5206 \\(\\displaystyle\\int _ {\\frac{\\pi}{3}}^{\\frac{\\pi}{2}} \\dfrac{2+\\sin x}{1+\\cos x} \\, dx\\) \u3092\u6c42\u3081\u3088.<\/p><\/li>\r\n<li><p><strong>(2)<\/strong>\u3000\u95a2\u6570 \\(y = \\dfrac{\\sqrt{x^2+1}}{x^2-3x}\\) \u306e\u5897\u6e1b, \u6975\u5024\u3092\u8abf\u3079, \u305d\u306e\u30b0\u30e9\u30d5\u306e\u6982\u5f62\u3092\u63cf\u3051. \u305f\u3060\u3057, \u30b0\u30e9\u30d5\u306e\u51f9\u51f8, \u5909\u66f2\u70b9\u306f\u8abf\u3079\u306a\u304f\u3066\u3088\u3044.<\/p><\/li>\r\n<\/ol>\r\n<hr \/>\r\n<!--more-->\r\n<h4>\u3010 \u89e3 \u7b54 \u3011<\/h4>\r\n<p><strong>(1)<\/strong><\/p>\r\n<p>\u6c42\u3081\u308b\u5b9a\u7a4d\u5206\u3092 \\(I\\) \u3068\u304a\u304f\u3068\r\n\\[\\begin{align}\r\nI & = \\displaystyle\\int _ {\\frac{\\pi}{3}}^{\\frac{\\pi}{2}} \\dfrac{1}{\\cos^2 \\frac{x}{2}} \\, dx -\\displaystyle\\int _ {\\frac{\\pi}{3}}^{\\frac{\\pi}{2}} \\dfrac{( 1+\\cos x )'}{1 +\\cos x} \\, dx \\\\\r\n& = \\left[ \\dfrac{1}{2} \\tan \\dfrac{x}{2} \\right] _ {\\frac{\\pi}{3}}^{\\frac{\\pi}{2}} -\\left[ \\log ( 1+\\cos x ) \\right] _ {\\frac{\\pi}{3}}^{\\frac{\\pi}{2}} \\\\\r\n& = \\dfrac{1}{2} \\left( 1 -\\dfrac{\\sqrt{3}}{3} \\right) -\\log \\dfrac{1}{1+\\frac{1}{2}} \\\\\r\n& =\\underline{\\dfrac{3 -\\sqrt{3}}{6} +\\log \\dfrac{3}{2}}\r\n\\end{align}\\]\r\n<p><strong>(2)<\/strong><\/p>\r\n<p>\u5b9a\u7fa9\u57df\u306f \\(x \\neq 0 , 3\\) .<br \/>\r\n\\[\\begin{align}\r\ny' & = \\dfrac{\\frac{2x}{2 \\sqrt{x^2+1}} \\cdot (x^2-3x) -\\sqrt{x^2+1} \\cdot (2x-3)}{(x^2-3x)^2} \\\\\r\n& = \\dfrac{(x^3-3x^2) -(2x^3-3x^2+2x-3)}{(x^2-3x)^2 \\sqrt{x^2+1}} \\\\\r\n& = -\\dfrac{(x-1)(x^2+x+3)}{(x^2-3x)^2 \\sqrt{x^2+1}}\r\n\\end{align}\\]\r\n\\(y' =0\\) \u3092\u89e3\u304f\u3068, \\(x=1\\) .<br \/>\r\n\u305d\u308c\u305e\u308c\u306e\u6975\u9650\u5024\u306f\r\n\\[\\begin{gather}\r\n\\displaystyle\\lim _ {x \\rightarrow \\pm \\infty} y \\rightarrow 0 \\ , \\ \\displaystyle\\lim _ {x \\rightarrow -0} y \\rightarrow \\infty \\ , \\ \\displaystyle\\lim _ {x \\rightarrow +0} y \\rightarrow -\\infty \\ , \\\\\r\n\\displaystyle\\lim _ {x \\rightarrow 3-0} y \\rightarrow -\\infty \\ , \\ \\displaystyle\\lim _ {x \\rightarrow 3+0} y \\rightarrow \\infty\r\n\\end{gather}\\]\r\n\u3057\u305f\u304c\u3063\u3066, \u5897\u6e1b\u8868\u306f\u4e0b\u306e\u3088\u3046\u306b\u306a\u308b.\r\n\\[\r\n\\begin{array}{c|ccccccccc} x & ( -\\infty ) & \\cdots & (0) & \\cdots & 1 & \\cdots & (3) & \\cdots & ( \\infty ) \\\\ \\hline y' & & + & & + & 0 & - & & - & \\\\ \\hline y & (0) & \\nearrow & ( \\pm \\infty ) & \\nearrow & -\\frac{\\sqrt{2}}{2} & \\searrow & ( \\mp \\infty ) & \\searrow & (0) \\end{array}\r\n\\]\r\n\u3088\u3063\u3066, \u30b0\u30e9\u30d5\u306e\u6982\u5f62\u306f\u4e0b\u56f3.<\/p>\r\n<img decoding=\"async\" src=\"\/\/www.roundown.net\/nyushi\/wp-content\/uploads\/yokokoku_r_2012_01_01.png\" alt=\"yokokoku_r_2012_01_01\" class=\"aligncenter size-full\" \/>\r\n","protected":false},"excerpt":{"rendered":"(1)\u3000\u5b9a\u7a4d\u5206 \\(\\displaystyle\\int _ {\\frac{\\pi}{3}}^{\\frac{\\pi}{2}} \\dfrac{2+\\sin x}{1+\\cos x} \\, dx\\) \u3092\u6c42\u3081\u3088. (2)\u3000\u95a2\u6570  &hellip; <a href=\"https:\/\/www.roundown.net\/nyushi\/ykr201201\/\">\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":[66],"tags":[68,9],"class_list":["post-423","post","type-post","status-publish","format-standard","hentry","category-yokokoku_r_2012","tag-68","tag-yokokoku_r"],"_links":{"self":[{"href":"https:\/\/www.roundown.net\/nyushi\/wp-json\/wp\/v2\/posts\/423","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=423"}],"version-history":[{"count":0,"href":"https:\/\/www.roundown.net\/nyushi\/wp-json\/wp\/v2\/posts\/423\/revisions"}],"wp:attachment":[{"href":"https:\/\/www.roundown.net\/nyushi\/wp-json\/wp\/v2\/media?parent=423"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/www.roundown.net\/nyushi\/wp-json\/wp\/v2\/categories?post=423"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/www.roundown.net\/nyushi\/wp-json\/wp\/v2\/tags?post=423"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}