{"id":2115,"date":"2011-12-18T17:34:32","date_gmt":"2011-12-18T08:34:32","guid":{"rendered":"http:\/\/njet.oops.jp\/wordpress\/?p=2115"},"modified":"2011-12-18T23:23:48","modified_gmt":"2011-12-18T14:23:48","slug":"%e6%a9%9f%e6%a2%b0%e3%81%ab%e8%b2%a0%e3%81%91%e3%82%8b","status":"publish","type":"post","link":"https:\/\/njet.oops.jp\/wordpress\/2011\/12\/18\/%e6%a9%9f%e6%a2%b0%e3%81%ab%e8%b2%a0%e3%81%91%e3%82%8b\/","title":{"rendered":"\u6a5f\u68b0\u306b\u8ca0\u3051\u308b"},"content":{"rendered":"<p>\u3068\u3042\u308b\u7a4d\u5206\u8a08\u7b97\u3002$\\int \\frac{\\cos^2 x}{\\sin^4 x}\\,dx$ \u3092\u6c42\u3081\u308b\u3053\u3068\u306b\u3002\u5206\u5b50\u3092$\\cos^2 x=1-\\sin^2 x$\u3068\u5909\u5f62\u3059\u308c\u3070\uff0c$I_n=\\int \\sin^n x\\,dx$ \u306b\u5e30\u7740\u3067\u304d\u305d\u3046\u3067\u3042\u308b\u3002<\/p>\n<p>$I_n$\u304c\u6e80\u305f\u3059\u6f38\u5316\u5f0f<br \/>\n\\[ nI_n=(n-1)I_{n-2}-\\sin^{n-1}x\\cos x \\]<br \/>\n\u306f$n$\u304c\u8ca0\u306e\u6574\u6570\u3067\u3082\u6210\u308a\u7acb\u3064\u306e\u3067\uff0c\u3053\u308c\u3092\u4f7f\u3048\u3070\uff0c$I_{-4}$\u306f<br \/>\n\\[ I_{-2}=\\int \\frac{1}{\\sin^2 x}\\,dx=-\\frac{\\cos x}{\\sin x} \\]<br \/>\n\u306b\u5e30\u7740\u3055\u308c\u308b\u304b\u3089\uff0c<br \/>\n\\begin{align*}<br \/>\n\\int \\frac{\\cos^2 x}{\\sin^4 x}\\,dx<br \/>\n&#038;= I_{-4}-I_{-2} \\\\<br \/>\n&#038;= \\frac13\\left( 2I_{-2}-\\frac{\\cos x}{\\sin^3 x} \\right)-I_{-2} \\\\<br \/>\n&#038;= \\frac13\\left( -I_{-2}-\\frac{\\cos x}{\\sin^3 x} \\right) \\\\<br \/>\n&#038;= \\frac13\\left( \\frac{\\cos x}{\\sin x}-\\frac{\\cos x}{\\sin^3 x} \\right) \\\\<br \/>\n&#038;= -\\frac13\\left( \\frac{\\cos x}{\\sin x} \\right)^3<br \/>\n\\end{align*}<br \/>\n\u3068\u306a\u308b\u3002<\/p>\n<p>\u5ff5\u306e\u305f\u3081\uff0cWolfram Alpha \u3067\u30c1\u30a7\u30c3\u30af\u3057\u3066\u307f\u305f\u3002\u306a\u3093\u3068\uff0c$u=\\cot x=\\frac{\\cos x}{\\sin x}$ \u3068\u5909\u6570\u5909\u63db\u3057\u3066\uff0c<br \/>\n\\begin{align*}<br \/>\n\\int \\frac{\\cos^2 x}{\\sin^4 x}\\,dx<br \/>\n&#038;= \\int \\frac{\\cos^2 x}{\\sin^2 x}\\cdot \\frac{1}{\\sin^2 x}\\,dx \\\\<br \/>\n&#038;= \\int u^2 \\,(-du) \\\\<br \/>\n&#038;= -\\frac13 u^3 \\\\<br \/>\n&#038;= -\\frac13\\left( \\frac{\\cos x}{\\sin x} \\right)^3<br \/>\n\\end{align*}<br \/>\n\u3068\u8a08\u7b97\u3057\u3066\u3044\u305f\u3002\u3048\u3048\uff0c\u305d\u3046\u3067\u3059\u3002\u3053\u306e\u65b9\u304c\u305a\u3063\u3068\u7c21\u5358\u3067\u3059\u3002\u8ca0\u3051\u307e\u3057\u305f\u3002\u3068\u3044\u3046\u304b\uff0c\u3082\u3046\u306d\uff0c\u7a4d\u5206\u8a08\u7b97\u3068\u304b\u6a5f\u68b0\u306e\u4ed5\u4e8b\u3060\u304b\u3089\u3055 \ud83d\ude09 \u3002\u3042\u30fc\u3042\u3002<\/p>\n","protected":false},"excerpt":{"rendered":"<p>\u3068\u3042\u308b\u7a4d\u5206\u8a08\u7b97\u3002$\\int \\frac{\\cos^2 x}{\\sin^4 x}\\,dx$ \u3092\u6c42\u3081\u308b\u3053\u3068\u306b\u3002\u5206\u5b50\u3092$\\cos^2 x=1-\\sin^2 x$\u3068\u5909\u5f62\u3059\u308c\u3070\uff0c$I_n=\\int \\sin^n x\\,dx$ \u306b\u5e30 [&hellip;]<\/p>\n","protected":false},"author":2,"featured_media":0,"comment_status":"open","ping_status":"open","sticky":false,"template":"","format":"standard","meta":{"footnotes":""},"categories":[10,5],"tags":[],"class_list":["post-2115","post","type-post","status-publish","format-standard","hentry","category-10","category-5"],"_links":{"self":[{"href":"https:\/\/njet.oops.jp\/wordpress\/wp-json\/wp\/v2\/posts\/2115","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/njet.oops.jp\/wordpress\/wp-json\/wp\/v2\/posts"}],"about":[{"href":"https:\/\/njet.oops.jp\/wordpress\/wp-json\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"https:\/\/njet.oops.jp\/wordpress\/wp-json\/wp\/v2\/users\/2"}],"replies":[{"embeddable":true,"href":"https:\/\/njet.oops.jp\/wordpress\/wp-json\/wp\/v2\/comments?post=2115"}],"version-history":[{"count":18,"href":"https:\/\/njet.oops.jp\/wordpress\/wp-json\/wp\/v2\/posts\/2115\/revisions"}],"predecessor-version":[{"id":2128,"href":"https:\/\/njet.oops.jp\/wordpress\/wp-json\/wp\/v2\/posts\/2115\/revisions\/2128"}],"wp:attachment":[{"href":"https:\/\/njet.oops.jp\/wordpress\/wp-json\/wp\/v2\/media?parent=2115"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/njet.oops.jp\/wordpress\/wp-json\/wp\/v2\/categories?post=2115"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/njet.oops.jp\/wordpress\/wp-json\/wp\/v2\/tags?post=2115"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}