{"id":138,"date":"2011-12-01T14:12:44","date_gmt":"2011-12-01T05:12:44","guid":{"rendered":"http:\/\/roundown.main.jp\/nyushi\/?p=138"},"modified":"2021-10-20T15:49:18","modified_gmt":"2021-10-20T06:49:18","slug":"ykr201001","status":"publish","type":"post","link":"https:\/\/www.roundown.net\/nyushi\/ykr201001\/","title":{"rendered":"\u6a2a\u56fd\u5927\u7406\u7cfb2010\uff1a\u7b2c1\u554f"},"content":{"rendered":"<ol>\r\n<li><p><strong>(1)<\/strong>\u3000\\(f(x)\\) \u3092\u9023\u7d9a\u95a2\u6570\u3068\u3059\u308b\u3068\u304d,\r\n\\[\r\n\\displaystyle\\int _ 0^\\pi x f \\left( \\sin x \\right) dx = \\dfrac{\\pi}{2} \\displaystyle\\int _ 0^\\pi f \\left( \\sin x \\right) dx\r\n\\]\r\n\u304c\u6210\u308a\u7acb\u3064\u3053\u3068\u3092\u793a\u305b.<\/p><\/li>\r\n<li><p><strong>(2)<\/strong>\u3000\u5b9a\u7a4d\u5206\r\n\\[\r\n\\displaystyle\\int _ 0^\\pi \\dfrac{x \\sin^3 x}{\\sin^2 x +8} \\, dx\r\n\\]\r\n\u306e\u5024\u3092\u6c42\u3081\u3088.<\/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>\\(I = \\displaystyle\\int _ 0^\\pi x f \\left( \\sin x \\right) dx\\) \u3068\u304a\u304f.<br \/>\r\n\\(x = \\pi -t\\) \u3068\u304a\u304f\u3068, \\(dx = -dt\\) \u3067\u3042\u308a\r\n\\[\r\n\\begin{array}{c|ccc} x & 0 & \\rightarrow & \\pi \\\\ \\hline t & \\pi & \\rightarrow & 0 \\end{array}\r\n\\]\r\n\u3057\u305f\u304c\u3063\u3066\r\n\\[\\begin{align}\r\nI & = -\\displaystyle\\int _ 0^\\pi ( \\pi -t ) f \\left( \\sin ( \\pi -t ) \\right) dt \\\\\r\n& = \\pi \\displaystyle\\int _ 0^\\pi f \\left( \\sin t \\right) dt -\\displaystyle\\int _ 0^\\pi t f \\left( \\sin t \\right) dt \\quad ( \\ \\text{\u2235} \\ \\sin ( \\pi -t ) = \\sin t \\ ) \\\\\r\n& = \\pi \\displaystyle\\int _ 0^\\pi f \\left( \\sin t \\right) dt -I \\\\\r\n\\text{\u2234} \\quad 2I & = \\pi \\displaystyle\\int _ 0^\\pi f \\left( \\sin x \\right) dx\r\n\\end{align}\\]\r\n\u3088\u3063\u3066\r\n\\[\r\nI = \\underline{\\dfrac{\\pi}{2} \\displaystyle\\int _ 0^\\pi f \\left( \\sin x \\right) dx}\r\n\\]\r\n<p><strong>(2)<\/strong><\/p>\r\n<p>\\(J = \\displaystyle\\int _ 0^\\pi \\dfrac{x \\sin^3 x}{\\sin^2 x +8} \\, dx\\) \u3068\u304a\u304f.<br \/>\r\n\\(f(x) = \\dfrac{x^3}{x^2+8} = x -\\dfrac{8x}{x^2+8}\\) \u3068\u304a\u304f\u3068, \u3053\u308c\u306f\u9023\u7d9a\u95a2\u6570\u306a\u306e\u3067, <strong>(1)<\/strong> \u306e\u7d50\u679c\u3092\u7528\u3044\u3066\r\n\\[\\begin{align}\r\nJ & = \\dfrac{\\pi}{2} \\displaystyle\\int _ 0^\\pi \\left( \\sin x -\\dfrac{8 \\sin x}{\\sin ^2 x +8} \\right) dx \\\\\r\n& = \\dfrac{\\pi}{2} \\displaystyle\\int _ 0^\\pi \\sin x dx -4 \\pi \\displaystyle\\int _ 0^\\pi \\dfrac{\\sin x}{9 -\\cos^2 x} \\, dx\r\n\\end{align}\\]\r\n\u3053\u3053\u3067\r\n\\[\\begin{align}\r\n\\displaystyle\\int _ 0^\\pi \\sin x \\, dx & = \\left[ -\\cos x \\right] _ 0^{\\pi} = 2 , \\\\\r\n\\displaystyle\\int _ 0^\\pi \\dfrac{\\sin x}{9 -\\cos^2 x} \\, dx & = -\\displaystyle\\int _ 0^{\\pi} \\dfrac{\\left( \\cos x \\right)'}{( 3 +\\cos x )( 3 -\\cos x )} \\, dx \\\\\r\n& = -\\dfrac{1}{6} \\displaystyle\\int _ 0^{\\pi} \\left( \\dfrac{1}{3 +\\cos x} + \\dfrac{1}{3 -\\cos x} \\right) \\left( \\cos x \\right)' \\, dx \\\\\r\n& = -\\dfrac{1}{6} \\left[ \\log \\dfrac{3 +\\cos x}{3 -\\cos x} \\right] _ 0^{\\pi} \\\\\r\n& = -\\dfrac{1}{6} \\left( \\log \\dfrac{1}{2} -\\log 2 \\right) \\\\\r\n& = \\dfrac{\\log 2}{3}\r\n\\end{align}\\]\r\n\u3057\u305f\u304c\u3063\u3066\r\n\\[\\begin{align}\r\nJ & = \\dfrac{\\pi}{2} \\cdot 2 -4 \\pi \\cdot \\dfrac{\\log 2}{3} \\\\\r\n& = \\underline{\\pi \\left( 1 -\\dfrac{4 \\log 2}{3} \\right)}\r\n\\end{align}\\]\r\n","protected":false},"excerpt":{"rendered":"(1)\u3000\\(f(x)\\) \u3092\u9023\u7d9a\u95a2\u6570\u3068\u3059\u308b\u3068\u304d, \\[ \\displaystyle\\int _ 0^\\pi x f \\left( \\sin x \\right) dx = \\dfrac{\\pi}{2} \\displayst &hellip; <a href=\"https:\/\/www.roundown.net\/nyushi\/ykr201001\/\">\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":[43],"tags":[14,9],"class_list":["post-138","post","type-post","status-publish","format-standard","hentry","category-yokokoku_r_2010","tag-14","tag-yokokoku_r"],"_links":{"self":[{"href":"https:\/\/www.roundown.net\/nyushi\/wp-json\/wp\/v2\/posts\/138","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=138"}],"version-history":[{"count":0,"href":"https:\/\/www.roundown.net\/nyushi\/wp-json\/wp\/v2\/posts\/138\/revisions"}],"wp:attachment":[{"href":"https:\/\/www.roundown.net\/nyushi\/wp-json\/wp\/v2\/media?parent=138"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/www.roundown.net\/nyushi\/wp-json\/wp\/v2\/categories?post=138"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/www.roundown.net\/nyushi\/wp-json\/wp\/v2\/tags?post=138"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}