<\/p>\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
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
\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":[],"aioseo_head":"\n\t\t\n\t\n\t\n\t\n\t\n\t\t\n\t\t\n\t\t\n\t\t\n\t\t\n\t\t\n\t\t\n\t\t\n\t\t\n\t\t\n\t\t\n\t\t