{"id":2146,"date":"2022-02-22T15:06:36","date_gmt":"2022-02-22T06:06:36","guid":{"rendered":"https:\/\/home.hirosaki-u.ac.jp\/relativity\/?page_id=2146"},"modified":"2024-05-22T14:28:42","modified_gmt":"2024-05-22T05:28:42","slug":"%e4%b8%8d%e5%ae%9a%e7%a9%8d%e5%88%86","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\/","title":{"rendered":"\u4e0d\u5b9a\u7a4d\u5206"},"content":{"rendered":"

\u5fae\u5206\u306e\u9006\u6f14\u7b97\u3068\u3057\u3066\u306e\u4e0d\u5b9a\u7a4d\u5206\u3002<\/p>\n

\u5fae\u5206\u306f\u4e0e\u3048\u3089\u308c\u305f\u95a2\u6570\u304b\u3089\uff0c\u305d\u306e\u5c0e\u95a2\u6570\u3092\u6c42\u3081\u308b\u3002\uff08\u4e0d\u5b9a\uff09\u7a4d\u5206\u3068\u306f\uff0c\u5c0e\u95a2\u6570\u304c\u4e0e\u3048\u3089\u308c\u305f\u3068\u304d\u306b\uff0c\u5fae\u5206\u3059\u308b\u524d\u306e\u3082\u3068\u306e\u95a2\u6570\u3092\u6c42\u3081\u308b\u3053\u3068\u3002\u305d\u306e\u610f\u5473\u3067\uff0c\u5fae\u5206\u306e\u9006\u6f14\u7b97\u3002<\/p>\n

\n

\u539f\u59cb\u95a2\u6570<\/h3>\n

\u95a2\u6570 \\(f(x)\\) \u304c\u4e0e\u3048\u3089\u308c\u305f\u3068\u304d\uff0c
\n$$ \\frac{d}{dx} F(x) = F'(x) = f(x)$$ \u3068\u306a\u308b \\(F(x)\\) \u3092 \\(f(x) \\) \u306e\u300c\u539f\u59cb\u95a2\u6570<\/strong><\/span>\u300d\u3068\u547c\u3076\u3002<\/p>\n

\u3053\u308c\u306f\uff0c\u95a2\u6570 \\(F(x)\\) \u304c\u4e0e\u3048\u3089\u308c\u305f\u3068\u304d\uff0c$$ \\frac{d}{dx} F(x) = F'(x) = f(x)$$ \u3068\u306a\u308b \\(f(x)\\) \u3092 \\(F(x) \\) \u306e\u300c\u5c0e\u95a2\u6570<\/strong><\/span>\u300d\u3068\u547c\u3093\u3060\u3053\u3068\u306e\u9006\u306e\u8a00\u3044\u56de\u3057\u306b\u306a\u3063\u3066\u3044\u308b\u3002<\/p>\n

\u4e0d\u5b9a\u7a4d\u5206\u3068\u7a4d\u5206\u5b9a\u6570<\/h3>\n

\u4eca\uff0c\u95a2\u6570 \\(f(x)\\) \u306e\u539f\u59cb\u95a2\u6570\u306e\u4e00\u3064 \\(F(x)\\) \u304c\u6c42\u3081\u3089\u308c\u305f\u3068\u3057\u3088\u3046\u3002\u305d\u306e\u3068\u304d\uff0c\\(F(x)\\) \u306b\u4efb\u610f\u5b9a\u6570 \\(C\\) \u3092\u8db3\u3057\u305f\u3082\u306e\u3082\u3084\u306f\u308a\u539f\u59cb\u95a2\u6570\u3067\u3042\u308b\u3002\u306a\u305c\u306a\u3089
\n$$(F(x) + C)’ = F'(x) + C’ = f(x), \\ \\ \\because C’ = 0$$<\/p>\n

\u3053\u306e\u3068\u304d\uff0c
\n$$\\int f(x) \\, dx = F(x) + C$$ \u3068\u66f8\u304d\uff0c\\(\\displaystyle \\int f(x) \\, dx \\) \u3092 \\(f(x)\\) \u306e\u300c\u4e0d\u5b9a\u7a4d\u5206<\/strong><\/span>\u300d\u3068\u3044\u3046\u3002\u307e\u305f\u4efb\u610f\u5b9a\u6570 \\(C\\) \u3092\u300c\u7a4d\u5206\u5b9a\u6570<\/strong><\/span>\u300d\u3068\u3044\u3046\u3002<\/p>\n

\\(f(x)\\) \u306e\u4e0d\u5b9a\u7a4d\u5206\u3092\u6c42\u3081\u308b\u3053\u3068\u3092\uff0c\u300c\u7a4d\u5206\u3059\u308b\u300d\u3068\u3044\u3046\u3053\u3068\u3082\u3042\u308b\u3002<\/p>\n

\u7a4d\u5206\u306e\u7dda\u5f62\u6027<\/h3>\n

\\(k\\) \u3092\u5b9a\u6570\u3068\u3059\u308b\u3068\uff0c\u4ee5\u4e0b\u306e\u3088\u3046\u306a\u95a2\u4fc2\u304c\u6210\u308a\u7acb\u3064\u3002
\n$$ \\int k f(x)\\, dx = k \\int f(x)\\, dx$$
\n$$\\int \\left\\{f(x) + g(x)\\right\\}\\, dx = \\int f(x)\\, dx + \\int g(x)\\, dx$$<\/p>\n

\u521d\u7b49\u95a2\u6570\u306e\u4e0d\u5b9a\u7a4d\u5206<\/h3>\n

\u3053\u308c\u307e\u3067\u8aac\u660e\u3057\u3066\u304d\u305f\u3088\u3046\u306b\uff0c\u3044\u308f\u3086\u308b\u521d\u7b49\u95a2\u6570\uff08\u3079\u304d\u95a2\u6570\uff0c\u6307\u6570\u95a2\u6570\uff0c\u5bfe\u6570\u95a2\u6570\uff0c\u4e09\u89d2\u95a2\u6570\u7b49\uff09\u306e\u5c0e\u95a2\u6570\u304c\u5b58\u5728\u3059\u308b\u3053\u3068\u306f\u308f\u304b\u3063\u3066\u3044\u308b\u306e\u3067\uff0c\u9006\u6f14\u7b97\u3068\u3057\u3066\u306e\u4e0d\u5b9a\u7a4d\u5206\u3082\u4ee5\u4e0b\u306e\u3088\u3046\u306b\u306a\u308b\u3053\u3068\u304c\u308f\u304b\u308b\u3002\uff08\u7a4d\u5206\u5b9a\u6570\u306f\u7701\u7565\u3002\uff09<\/p>\n

\\begin{eqnarray}
\n(x^{p+1})’ = (p+1) \\, x^{p} \\ \\ &\\Longleftrightarrow& \\ \\ \\int x^{p}\\, dx = \\frac{1}{p+1} x^{p+1}\\ \\ (p \\neq -1)\\\\
\n(e^x)’ = e^x \\ \\ &\\Longleftrightarrow& \\ \\ \\int e^x \\, dx = e^x\\\\
\n(\\log |x|)’ = \\frac{1}{x} \\ \\ &\\Longleftrightarrow& \\ \\ \\int\\frac{1}{x} \\, dx = \\log|x|\\\\
\n(\\sin x)’ = \\cos x \\ \\ &\\Longleftrightarrow& \\ \\ \\int \\cos x \\, dx = \\sin x\\\\
\n(\\cos x)’ = -\\sin x \\ \\ &\\Longleftrightarrow& \\ \\ \\int \\sin x \\, dx = -\\cos x \\\\
\n(\\tan x)’ = \\frac{1}{\\cos^2 x} \\ \\ &\\Longleftrightarrow& \\ \\ \\int \\frac{1}{\\cos^2 x}\\, dx = \\tan x\\\\
\n(\\sin^{-1} x)’ = \\frac{1}{\\sqrt{1-x^2}} \\ \\ &\\Longleftrightarrow& \\ \\ \\int \\frac{1}{\\sqrt{1-x^2}}\\, dx = \\sin^{-1} x\\\\
\n(\\cos^{-1} x)’ = -\\frac{1}{\\sqrt{1-x^2}}\\ \\ &\\Longleftrightarrow& \\ \\ \\int \\left\\{-\\frac{1}{\\sqrt{1-x^2}}\\right\\}\\, dx = \\cos^{-1} x\\\\
\n(\\tan^{-1} x)’ =\u00a0 \\frac{1}{1+x^2}\\ \\ &\\Longleftrightarrow& \\ \\ \\int \\frac{1}{1+x^2}\\, dx = \\tan^{-1} x \\\\
\n(\\sinh x)’ = \\cosh x\\ \\ &\\Longleftrightarrow& \\ \\ \\int \\cosh x \\, dx = \\sinh x \\\\
\n(\\cosh x)’ = \\sinh x\\ \\ &\\Longleftrightarrow& \\ \\ \\int \\sinh x\\, dx = \\cosh x \\\\
\n(\\tanh x)’ = \\frac{1}{\\cosh^2 x}\\ \\ &\\Longleftrightarrow& \\ \\ \\int \\frac{1}{\\cosh^2 x}\\, dx = \\tanh x \\\\
\n(\\sinh^{-1} x)’ =\u00a0 \\frac{1}{\\sqrt{x^2 + 1}}\\ \\ &\\Longleftrightarrow& \\ \\ \\int \\frac{1}{\\sqrt{x^2 + 1}}\\, dx = \\sinh^{-1} x \\\\
\n(\\cosh^{-1} x)’ =\u00a0 \\frac{1}{\\sqrt{x^2-1}}\\ \\ &\\Longleftrightarrow& \\ \\ \\int \\frac{1}{\\sqrt{x^2-1}}\\, dx = \\cosh^{-1} x \\\\
\n(\\tanh^{-1} x)’ =\u00a0 \\frac{1}{1-x^2}\\ \\ &\\Longleftrightarrow& \\ \\ \\int \\frac{1}{1-x^2}\\, dx = \\tanh^{-1} x
\n\\end{eqnarray}<\/p>\n

\u7d20\u6734\u306a\u7591\u554f\uff1a$\\cos^{-1} x$ \u304b $-\\sin^{-1} x$ \u304b<\/h3>\n

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

\u306a\u3089<\/p>\n

$$\\int \\left\\{-\\frac{1}{\\sqrt{1-x^2}}\\right\\}\\, dx = -\\int \\frac{1}{\\sqrt{1-x^2}}\\, dx = \\cos^{-1} x$$<\/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\u3068\u7591\u554f\u3092\u3082\u3064\u3042\u306a\u305f\u3078\u3002\u5225\u30da\u30fc\u30b8\u3092\u53c2\u7167\u306e\u3053\u3068\u3002<\/p>\n","protected":false},"excerpt":{"rendered":"

\u5fae\u5206\u306e\u9006\u6f14\u7b97\u3068\u3057\u3066\u306e\u4e0d\u5b9a\u7a4d\u5206\u3002<\/p>\n

\u5fae\u5206\u306f\u4e0e\u3048\u3089\u308c\u305f\u95a2\u6570\u304b\u3089\uff0c\u305d\u306e\u5c0e\u95a2\u6570\u3092\u6c42\u3081\u308b\u3002\uff08\u4e0d\u5b9a\uff09\u7a4d\u5206\u3068\u306f\uff0c\u5c0e\u95a2\u6570\u304c\u4e0e\u3048\u3089\u308c\u305f\u3068\u304d\u306b\uff0c\u5fae\u5206\u3059\u308b\u524d\u306e\u3082\u3068\u306e\u95a2\u6570\u3092\u6c42\u3081\u308b\u3053\u3068\u3002\u305d\u306e\u610f\u5473\u3067\uff0c\u5fae\u5206\u306e\u9006\u6f14\u7b97\u3002<\/p>

\u7d9a\u304d\u3092\u8aad\u3080<\/a><\/p>\n","protected":false},"author":33,"featured_media":0,"parent":2068,"menu_order":21,"comment_status":"closed","ping_status":"closed","template":"","meta":{"inline_featured_image":false,"footnotes":""},"class_list":["post-2146","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\/2146","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=2146"}],"version-history":[{"count":9,"href":"https:\/\/home.hirosaki-u.ac.jp\/relativity\/wp-json\/wp\/v2\/pages\/2146\/revisions"}],"predecessor-version":[{"id":8768,"href":"https:\/\/home.hirosaki-u.ac.jp\/relativity\/wp-json\/wp\/v2\/pages\/2146\/revisions\/8768"}],"up":[{"embeddable":true,"href":"https:\/\/home.hirosaki-u.ac.jp\/relativity\/wp-json\/wp\/v2\/pages\/2068"}],"wp:attachment":[{"href":"https:\/\/home.hirosaki-u.ac.jp\/relativity\/wp-json\/wp\/v2\/media?parent=2146"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}