\u66f2\u7dda \\(y = \\log ( 1 +\\cos x )\\) \u306e \\(0 \\leqq x \\leqq \\dfrac{\\pi}{2}\\) \u306e\u90e8\u5206\u306e\u9577\u3055\u3092\u6c42\u3081\u3088.<\/p>\r\n
\\[\r\ny' = -\\dfrac{\\sin x}{1 +\\cos x}\r\n\\]\r\n\u306a\u306e\u3067\r\n\\[\\begin{align}\r\n\\sqrt{1 +\\left\\{ y' \\right\\}^2} & = \\sqrt{\\dfrac{2 +2 \\cos x}{( 1 +\\cos x )^2}} \\\\\r\n& = \\dfrac{1}{\\sqrt{\\cos^2 \\dfrac{x}{2}}} = \\dfrac{1}{\\cos \\dfrac{x}{2}}\\quad \\left( \\text{\u2235} \\cos \\dfrac{x}{2} \\gt 0 \\right)\r\n\\end{align}\\]\r\n\u3088\u3063\u3066, \u6c42\u3081\u308b\u9577\u3055 \\(L\\) \u306f\r\n\\[\\begin{align}\r\nL & = \\displaystyle\\int _ {0}^{\\frac{\\pi}{2}} \\dfrac{1}{\\cos \\dfrac{x}{2}} \\, dx = \\displaystyle\\int _ {0}^{\\frac{\\pi}{2}} \\dfrac{\\cos \\dfrac{x}{2}}{1 -\\sin^2 \\dfrac{x}{2}} \\, dx \\\\\r\n& = \\displaystyle\\int _ {0}^{\\frac{\\pi}{2}} \\left\\{ \\dfrac{\\left( 1 +\\sin \\dfrac{x}{2} \\right)'}{1 +\\sin \\dfrac{x}{2}} -\\dfrac{\\left( 1 -\\sin \\dfrac{x}{2} \\right)'}{1 -\\sin \\dfrac{x}{2}} \\right\\} \\, dx \\\\\r\n& = \\left[ \\log \\dfrac{1 +\\sin \\dfrac{x}{2}}{1 -\\sin \\dfrac{x}{2}} \\right]_ {0}^{\\frac{\\pi}{2}} \\\\\r\n& = \\log \\dfrac{\\sqrt{2} +1}{\\sqrt{2} -1} = \\underline{2 \\log \\left( \\sqrt{2} +1 \\right)}\r\n\\end{align}\\]\r\n","protected":false},"excerpt":{"rendered":"\u66f2\u7dda \\(y = \\log ( 1 +\\cos x )\\) \u306e \\(0 \\leqq x \\leqq \\dfrac{\\pi}{2}\\) \u306e\u90e8\u5206\u306e\u9577\u3055\u3092\u6c42\u3081\u3088.","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":[171],"tags":[140,165],"class_list":["post-1972","post","type-post","status-publish","format-standard","hentry","category-kyoto_r_2021","tag-kyoto_r","tag-165"],"_links":{"self":[{"href":"https:\/\/www.roundown.net\/nyushi\/wp-json\/wp\/v2\/posts\/1972","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=1972"}],"version-history":[{"count":0,"href":"https:\/\/www.roundown.net\/nyushi\/wp-json\/wp\/v2\/posts\/1972\/revisions"}],"wp:attachment":[{"href":"https:\/\/www.roundown.net\/nyushi\/wp-json\/wp\/v2\/media?parent=1972"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/www.roundown.net\/nyushi\/wp-json\/wp\/v2\/categories?post=1972"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/www.roundown.net\/nyushi\/wp-json\/wp\/v2\/tags?post=1972"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}