{"id":2167,"date":"2022-02-22T17:03:22","date_gmt":"2022-02-22T08:03:22","guid":{"rendered":"https:\/\/home.hirosaki-u.ac.jp\/relativity\/?page_id=2167"},"modified":"2024-04-17T10:30:15","modified_gmt":"2024-04-17T01:30:15","slug":"%e5%8f%82%e8%80%83%ef%bc%9a%e7%b4%a0%e6%9c%b4%e3%81%aa%e7%96%91%e5%95%8f%e3%81%b8%e3%81%ae%e8%a7%a3%e7%ad%94%e4%be%8b","status":"publish","type":"page","link":"https:\/\/home.hirosaki-u.ac.jp\/relativity\/%e7%90%86%e5%b7%a5%e7%b3%bb%e3%81%ae%e6%95%b0%e5%ad%a6b\/%e4%b8%8d%e5%ae%9a%e7%a9%8d%e5%88%86\/%e5%8f%82%e8%80%83%ef%bc%9a%e7%b4%a0%e6%9c%b4%e3%81%aa%e7%96%91%e5%95%8f%e3%81%b8%e3%81%ae%e8%a7%a3%e7%ad%94%e4%be%8b\/","title":{"rendered":"\u53c2\u8003\uff1a\u7d20\u6734\u306a\u7591\u554f\u3078\u306e\u89e3\u7b54\u4f8b"},"content":{"rendered":"

<\/p>\n

\u7d20\u6734\u306a\u7591\u554f<\/h3>\n

$$\\int \\frac{1}{\\sqrt{1-x^2}}\\, dx = \\sin^{-1} x $$<\/p>\n

\u306a\u3089<\/p>\n

$$-\\int \\frac{1}{\\sqrt{1-x^2}}\\, dx = \\cos^{-1} x$$<\/p>\n

\u3058\u3083\u3042\u306a\u304f\u3066<\/p>\n

$$-\\int \\frac{1}{\\sqrt{1-x^2}}\\, dx = -\\sin^{-1} x $$<\/p>\n

\u306a\u3093\u3058\u3083\u3042\u306a\u3044\u306e\uff1f\u3000\u305d\u308c\u3068\u3082 $\\cos^{-1} x$ \u3068 $ -\\sin^{-1} x$ \u306f\u540c\u3058\u306a\u306e\uff1f<\/p>\n

\u89e3\u7b54\u4f8b<\/h3>\n

1. \u307e\u305a\uff0c\u4e0a\u8a18\u306e\u7a4d\u5206\u306f\u4e0d\u5b9a\u7a4d\u5206\u306a\u306e\u3067\uff0c\u7a4d\u5206\u5b9a\u6570\u3082\u7701\u7565\u305b\u305a\u306b\u66f8\u304f\u3068<\/p>\n

\\begin{eqnarray}
\n\\int \\frac{1}{\\sqrt{1-x^2}}\\, dx &=& \\cos^{-1} x + C_1 \\\\
\n&=& -\\sin^{-1} x + C_2
\n\\end{eqnarray}<\/p>\n

\u3068\u3044\u3046\u3053\u3068\u3002<\/p>\n

2. $\\cos^{-1} x$ \u3068 $ -\\sin^{-1} x$ \u306f\u7b49\u3057\u3044\u308f\u3051\u3067\u306f\u306a\u3044\u304c\uff0c\u4ee5\u4e0b\u306e\u3088\u3046\u306b\u5b9a\u6570\u5206\u3060\u3051\u7570\u306a\u308b\u3002<\/p>\n

$$ \\sin^{-1} x + \\cos^{-1} x = \\frac{\\pi}{2}$$<\/p>\n

\u5148\u306b\u4e0a\u8a18\u306e\u95a2\u4fc2\u3092\u8a3c\u660e\u3057\u3066\u304a\u3053\u3046\u3002<\/p>\n

$$x = \\cos y = \\sin \\left( \\frac{\\pi}{2} -y\\right)$$<\/p>\n

\u304b\u3089\uff0c$x = \\cos y$ \u3092 $y$ \u306b\u3064\u3044\u3066\u89e3\u304f\u3068<\/p>\n

$$y = \\cos^{-1} x$$<\/p>\n

\u4e00\u65b9\uff0c$\\displaystyle x =\\sin \\left( \\frac{\\pi}{2} -y\\right)$ \u3092 $y$ \u306b\u3064\u3044\u3066\u89e3\u304f\u3068<\/p>\n

$$ \\frac{\\pi}{2} -y = \\sin^{-1} x$$<\/p>\n

$$\\therefore\\ \\\u00a0 \\sin^{-1} x + y = \\sin^{-1} x + \\cos^{-1} x = \\frac{\\pi}{2}$$<\/p>\n

1. \u3068 2. \u304b\u3089\uff0c$\\cos^{-1} x$ \u3068 $ -\\sin^{-1} x$ \u306e\u9055\u3044\u306f\u7a4d\u5206\u5b9a\u6570\u306b\u5438\u53ce\u3055\u308c\u308b\u3053\u3068\u304c\u308f\u304b\u308b\u3002\u3042\u304b\u3089\u3055\u307e\u306b\u66f8\u304f\u3068\uff0c$\\displaystyle C_2 = \\frac{\\pi}{2} + C_1$ \u3068\u304a\u3051\u3070\uff0c<\/p>\n

$$-\\sin^{-1} x + C_2 = -\\sin^{-1} x + \\frac{\\pi}{2} + C_1 = \\cos^{-1} x+ C_1$$<\/p>\n

\u3068\u306a\u308b\u3002<\/p>\n

\u5fdc\u7528\u4f8b\uff1a\u5fae\u5206\u65b9\u7a0b\u5f0f<\/h3>\n

\u9006\u4e09\u89d2\u95a2\u6570\u304c\u51fa\u3066\u304f\u308b\u4f8b\u3068\u3057\u3066\uff0c\u4ee5\u4e0b\u306e\u3088\u3046\u306a\u5fae\u5206\u65b9\u7a0b\u5f0f\u3092\u8003\u3048\u308b\u3002\uff081\u5e74\u751f\u3067\u306f\u5fae\u5206\u65b9\u7a0b\u5f0f\u306f\u307e\u3060\u7fd2\u308f\u306a\u3044\u3051\u3069\uff0c\u53c2\u8003\u307e\u3067\u306b\u3002\uff09<\/p>\n

$$\\left( \\frac{dy}{dx}\\right)^2 = 1 -y^2$$<\/p>\n

\u3053\u308c\u306f\uff0c\u4ee5\u4e0b\u306e\u3088\u3046\u306b\u5909\u6570\u5206\u96e2\u304c\u3067\u304d\u3066<\/p>\n

$$dx = \\pm \\frac{dy}{\\sqrt{1 -y^2}}$$<\/p>\n

\u5fa9\u53f7\u90e8\u5206\u304c $+$ \u306e\u3068\u304d\u306f\u4ee5\u4e0b\u306e\u3088\u3046\u306b\u7a4d\u5206\u3067\u304d\u3066<\/p>\n

$$x = + \\int \\frac{dy}{\\sqrt{1 -y^2}} = \\sin^{-1} y$$<\/p>\n

\u5fa9\u53f7\u90e8\u5206\u304c $-$ \u306e\u3068\u304d\u306f<\/p>\n

$$x = -\\int \\frac{dy}{\\sqrt{1 -y^2}} = \\cos^{-1} y$$<\/p>\n

\u3055\u3066\uff0c\u3069\u3063\u3061\u3092\u7b54\u3048\u306b\u66f8\u3051\u3070\u3044\u3044\u3060\u308d\u3046\uff1f\u3068\u60a9\u3080\u3042\u306a\u305f\uff0c\u7a4d\u5206\u5b9a\u6570\u3092\uff08\u5de6\u8fba\u306b\uff09\u3064\u3051\u3066<\/p>\n

$$x + C = + \\int \\frac{dy}{\\sqrt{1 -y^2} }= \\sin^{-1} y$$<\/p>\n

\u3057\u3066\u304a\u3051\u3070\uff0c\u5fa9\u53f7\u90e8\u5206 $\\pm$ \u3069\u3061\u3089\u3082\u542b\u3093\u3060\u7b54\u3048\u306b\u306a\u308a\u307e\u3059\u3002$y$ \u306b\u3064\u3044\u3066\u89e3\u3051\u3070<\/p>\n

$$ y = \\sin( x + C)$$<\/p>\n

 <\/p>\n

 <\/p>\n","protected":false},"excerpt":{"rendered":"","protected":false},"author":33,"featured_media":0,"parent":2146,"menu_order":1,"comment_status":"closed","ping_status":"closed","template":"","meta":{"inline_featured_image":false,"footnotes":""},"class_list":["post-2167","page","type-page","status-publish","hentry","nodate","item-wrap"],"aioseo_notices":[],"_links":{"self":[{"href":"https:\/\/home.hirosaki-u.ac.jp\/relativity\/wp-json\/wp\/v2\/pages\/2167","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/home.hirosaki-u.ac.jp\/relativity\/wp-json\/wp\/v2\/pages"}],"about":[{"href":"https:\/\/home.hirosaki-u.ac.jp\/relativity\/wp-json\/wp\/v2\/types\/page"}],"author":[{"embeddable":true,"href":"https:\/\/home.hirosaki-u.ac.jp\/relativity\/wp-json\/wp\/v2\/users\/33"}],"replies":[{"embeddable":true,"href":"https:\/\/home.hirosaki-u.ac.jp\/relativity\/wp-json\/wp\/v2\/comments?post=2167"}],"version-history":[{"count":9,"href":"https:\/\/home.hirosaki-u.ac.jp\/relativity\/wp-json\/wp\/v2\/pages\/2167\/revisions"}],"predecessor-version":[{"id":8427,"href":"https:\/\/home.hirosaki-u.ac.jp\/relativity\/wp-json\/wp\/v2\/pages\/2167\/revisions\/8427"}],"up":[{"embeddable":true,"href":"https:\/\/home.hirosaki-u.ac.jp\/relativity\/wp-json\/wp\/v2\/pages\/2146"}],"wp:attachment":[{"href":"https:\/\/home.hirosaki-u.ac.jp\/relativity\/wp-json\/wp\/v2\/media?parent=2167"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}