{"id":2153,"date":"2022-02-22T15:10:07","date_gmt":"2022-02-22T06:10:07","guid":{"rendered":"https:\/\/home.hirosaki-u.ac.jp\/relativity\/?page_id=2153"},"modified":"2024-04-17T10:31:48","modified_gmt":"2024-04-17T01:31:48","slug":"%e7%bd%ae%e6%8f%9b%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\/%e7%bd%ae%e6%8f%9b%e7%a9%8d%e5%88%86\/","title":{"rendered":"\u7f6e\u63db\u7a4d\u5206"},"content":{"rendered":"

\u7a4d\u5206\u5909\u6570 \\(x\\) \u3092\u3046\u307e\u304f\u5909\u63db\u3059\u308b\u3068\uff0c\u7a4d\u5206\u306e\u8a08\u7b97\u304c\u7c21\u5358\u306b\u306a\u308b\u5834\u5408\u304c\u3042\u308b\u3002<\/p>\n

\u3064\u307e\u308a\uff0c\u88ab\u7a4d\u5206\u95a2\u6570 \\(f(x)\\) \u306b\u304a\u3044\u3066\uff0c\\( x = \\varphi(t)\\) \u3068\u7f6e\u3044\u3066\uff0c\\(x\\) \u306b\u95a2\u3059\u308b\u7a4d\u5206\u3092\u65b0\u3057\u3044\u5909\u6570 \\(t\\) \u306b\u95a2\u3059\u308b\u7a4d\u5206\u306b\u7f6e\u304d\u63db\u3048\u308b\u3002<\/p>\n

\u3053\u306e\u3068\u304d\uff0c\u88ab\u7a4d\u5206\u95a2\u6570\u3092\u65b0\u305f\u306a\u5909\u6570 \\(t\\) \u3067\u8868\u3057\uff0c\u307e\u305f\u5fae\u5206 \\(dx\\) \u3092 \\(\\displaystyle \\frac{dx}{dt} dt = \\frac{d\\varphi}{dt} dt = \\varphi'(t) dt \\) \u3067\u7f6e\u304d\u63db\u3048\uff0c<\/p>\n

$$\\int f(x)\\,dx = \\int f[\\varphi(t)] \\varphi'(t) \\,dt$$<\/p>\n

\u5b9a\u7a4d\u5206\u306e\u5834\u5408\u306b\u306f \\(x\\) \u306e\u7a4d\u5206\u7bc4\u56f2 \\(a, b\\) \u3082 \\(a = \\varphi(\\alpha), b = \\varphi(\\beta)\\) \u3068\u306a\u308b \\(t\\) \u306e\u7a4d\u5206\u7bc4\u56f2 \\(\\alpha, \\beta\\) \u306b\u7f6e\u304d\u63db\u3048\u3066\uff0c\u4ee5\u4e0b\u306e\u3088\u3046\u306b\u306a\u308b\u3002
\n$$\\int_a^b f(x)\\,dx = \\int_{\\alpha}^{\\beta} f[\\varphi(t)] \\varphi'(t) \\,dt$$<\/p>\n

… \u3068\u3044\u3046\u3053\u3068\u306b\u306a\u308b\u304c\uff0c\u73fe\u5b9f\u7684\u306b\u306f \\( t = g(x)\\) \u3068\u3057\u3066 \\(x\\) \u306e\u95a2\u6570\u3068\u3057\u3066\u65b0\u305f\u306a\u5909\u6570 \\(t\\) \u3092\u5b9a\u7fa9\u3059\u308b\u3053\u3068\u306b\u306a\u308b\u3060\u308d\u3046\u3002\u307e\u305f\uff0c\u4e0d\u5b9a\u7a4d\u5206\u306e\u5834\u5408\u306f\u6700\u7d42\u7684\u306a\u7d50\u679c\u306f\u5143\u306e\u7a4d\u5206\u5909\u6570 \\(x\\) \u3067\u8868\u3059\u3053\u3068\u306b\u306a\u308b\u3002<\/p>\n

\u4f8b\u984c 1<\/h3>\n

\\(\\displaystyle \\int \\frac{\\log x}{x} dx\\)<\/p>\n

\u3053\u308c\u306f\uff0c\\(x = e^t\\) \u3068\u7f6e\u304f… \u3068\u3044\u3046\u3088\u308a\u306f\uff0c\\( \\log x \\equiv t\\) \u3068\u7f6e\u304f\u3068\u3059\u308b\u306e\u304c\u73fe\u5b9f\u7684\u306a\u767a\u60f3\u3067\u3042\u308d\u3046\u3002\u305d\u3046\u3059\u308b\u3068\uff0c\\(\\displaystyle\u00a0 \\frac{1}{x} dx = dt\\) \u3067\u3042\u308b\u304b\u3089\uff0c
\n\\begin{eqnarray}
\n\\int \\frac{\\log x}{x} dx &=& \\int t\\, dt \\\\
\n&=& \\frac{t^2}{2} + C\\\\
\n&=& \\frac{(\\log x)^2}{2} + C
\n\\end{eqnarray}<\/p>\n

\u4f8b\u984c 2<\/h3>\n

\\(\\displaystyle \\int \\sin ax\\, \\cos ax\\, dx\\)<\/p>\n

\u305f\u3068\u3048\u3070\uff0c\\( \\sin ax \\equiv t\\) \u3068\u304a\u304f\u3068<\/p>\n

\\begin{eqnarray}
\n\\cos ax\\ a dx &=& dt \\\\
\n\\therefore\\ \\ \\cos ax\\ dx &=& \\frac{dt}{a} \\\\
\n\\therefore\\ \\ \\int \\sin ax\\, \\cos ax\\, dx &=& \\int t \\ \\frac{dt}{a} \\\\
\n&=& \\frac{t^2}{2 a} + C \\\\
\n&=& \\frac{\\sin^2 a x}{2 a} + C
\n\\end{eqnarray}<\/p>\n

\u3042\u308b\u3044\u306f\uff0c\\( \\cos ax \\equiv t\\) \u3068\u304a\u304f\u3068<\/p>\n

\\begin{eqnarray}
\n-\\sin ax\\ a dx &=& dt \\\\
\n\\therefore\\ \\ \\sin ax\\ dx &=& -\\frac{dt}{a} \\\\
\n\\therefore\\ \\ \\int \\sin ax\\, \\cos ax\\, dx &=& -\\int t \\ \\frac{dt}{a} \\\\
\n&=& -\\frac{t^2}{2 a} + C \\\\
\n&=& -\\frac{\\cos^2 a x}{2 a} + C
\n\\end{eqnarray}<\/p>\n

\u3042\u308b\u3044\u306f\u307e\u305f\uff0c\u500d\u89d2\u306e\u516c\u5f0f\u3092\u4f7f\u3046\u3068<\/p>\n

\\begin{eqnarray}
\n\\int \\sin ax\\, \\cos ax\\\u00a0 dx &=& \\frac{1}{2} \\int \\sin (2 a x) \\ dx \\\\
\n&=& \\frac{1}{4a} \\int \\sin (2 a x) \\cdot 2 a dx \\\\
\n&& \\quad (2 a x \\equiv t) \\\\
\n&=& \\frac{1}{4a} \\int \\sin t \\ dt \\\\
\n&=& -\\frac{\\cos t}{4a} + C \\\\
\n&=& -\\frac{\\cos 2 a x}{4a} + C
\n\\end{eqnarray}<\/p>\n

\u3068\u306a\u308a\uff0c1\u3064\u306e\u4e0d\u5b9a\u7a4d\u5206\u306b\u5bfe\u3057\u3066\u898b\u304b\u3051\u4e0a\uff0c3\u901a\u308a\u306e\u7b54\u3048\u304c\u3067\u3066\u304f\u308b\u3002\u3053\u308c\u3089\u304c\uff08\u7a4d\u5206\u5b9a\u6570\u306e\u4e0d\u5b9a\u6027\u3092\u8003\u616e\u3059\u308c\u3070\uff09\u540c\u7b49\u3067\u3042\u308b\u3053\u3068\u3092\u78ba\u8a8d\u3057\u3066\u304a\u304f\u3053\u3068\u3002<\/p>\n

Maxima \u3084 Python \u306e SymPy \u3092\u4f7f\u3063\u3066\u30b3\u30f3\u30d4\u30e5\u30fc\u30bf\u3067\u7a4d\u5206\u3055\u305b\u308b\u3068\uff0c\u4ee5\u4e0b\u306e\u3088\u3046\u306b\u4e00\u898b\u5225\u306e\u7b54\u3048\u304c\u51fa\u3066\u304f\u308b\u306e\u3067\uff0c\u305d\u3093\u306a\u6642\u3082\u5c11\u3057\u3082\u9a12\u304c\u305a\uff0c\u305d\u306e\u540c\u7b49\u6027\u3092\u793a\u3059\u306e\u304c\uff08\u6a5f\u68b0\u3067\u306f\u306a\u304f\uff09\u4eba\u985e\u306e\u5f79\u5272\u306a\u3093\u3067\u3059\u3088\u3002<\/p>\n

 <\/p>\n

\n

 <\/p>\n

\n
\n
\n

\u30b3\u30f3\u30d4\u30e5\u30fc\u30bf\u4ee3\u6570\u30b7\u30b9\u30c6\u30e0\u306e\u4e00\u3064\uff0cMaxima \u3067\u3053\u306e\u7a4d\u5206\u3092\u3055\u305b\u308b\u3068…<\/p>\n

Maxima \u3067 $\\displaystyle \\int \\sin(a x)\\,\\cos(a x)\\ dx$ \u306e\u7a4d\u5206<\/h5>\n<\/div>\n<\/div>\n<\/div>\n
\n
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In\u00a0[1]:<\/div>\n
\n
\n
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'<\/span>integrate<\/span>(<\/span>sin<\/span>(<\/span>a<\/span>*<\/span>x<\/span>)<\/span>*<\/span>cos<\/span>(<\/span>a<\/span>*<\/span>x<\/span>)<\/span>, x<\/span>)<\/span> =<\/span> \r\n integrate<\/span>(<\/span>sin<\/span>(<\/span>a<\/span>*<\/span>x<\/span>)<\/span>*<\/span>cos<\/span>(<\/span>a<\/span>*<\/span>x<\/span>)<\/span>, x<\/span>)<\/span>;\r\n<\/pre>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n
\n
\n
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Out[1]:<\/div>\n
\\[\\tag{${\\it \\%o}_{1}$}\\int {\\cos \\left(a\\,x\\right)\\,\\sin \\left(a\\,x\\right)}{\\;dx}=-\\frac{\\cos ^2\\left(a\\,x\\right)}{2\\,a}\\]<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n
 <\/p>\n
\n
 <\/p>\n
\u4e00\u65b9\uff0cPython \u306e SymPy \u3067\u306f\uff0c<\/p>\n
\n
\n
\n
Python \u306e SymPy \u3067 $\\displaystyle \\int \\sin(a x)\\,\\cos(a x)\\ dx$ \u306e\u7a4d\u5206<\/h5>\n<\/div>\n<\/div>\n<\/div>\n
\n
\n
In\u00a0[1]:<\/div>\n
\n
\n
\n
from<\/span> sympy<\/span> import<\/span> *<\/span>\r\nfrom<\/span> sympy.abc<\/span> import<\/span> *<\/span>\r\n<\/pre>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n
\n
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In\u00a0[2]:<\/div>\n
\n
\n
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Eq<\/span>(<\/span>Integral<\/span>(<\/span>sin<\/span>(<\/span>a<\/span>*<\/span>x<\/span>)<\/span>*<\/span>cos<\/span>(<\/span>a<\/span>*<\/span>x<\/span>),<\/span> x<\/span>),<\/span> \r\n   integrate<\/span>(<\/span>sin<\/span>(<\/span>a<\/span>*<\/span>x<\/span>)<\/span>*<\/span>cos<\/span>(<\/span>a<\/span>*<\/span>x<\/span>),<\/span> x<\/span>))<\/span>\r\n<\/pre>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n
\n
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Out[2]:<\/div>\n
$\\displaystyle \\int \\sin{\\left(a x \\right)} \\cos{\\left(a x \\right)}\\, dx = \\begin{cases} \\frac{\\sin^{2}{\\left(a x \\right)}}{2 a} & \\text{for}\\: a \\neq 0 \\\\0 & \\text{otherwise} \\end{cases}$<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n","protected":false},"excerpt":{"rendered":"
\u7a4d\u5206\u5909\u6570 \\(x\\) \u3092\u3046\u307e\u304f\u5909\u63db\u3059\u308b\u3068\uff0c\u7a4d\u5206\u306e\u8a08\u7b97\u304c\u7c21\u5358\u306b\u306a\u308b\u5834\u5408\u304c\u3042\u308b\u3002<\/p>
\u7d9a\u304d\u3092\u8aad\u3080<\/a><\/p>\n","protected":false},"author":33,"featured_media":0,"parent":2068,"menu_order":23,"comment_status":"closed","ping_status":"closed","template":"","meta":{"inline_featured_image":false,"footnotes":""},"class_list":["post-2153","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\/2153","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=2153"}],"version-history":[{"count":12,"href":"https:\/\/home.hirosaki-u.ac.jp\/relativity\/wp-json\/wp\/v2\/pages\/2153\/revisions"}],"predecessor-version":[{"id":8429,"href":"https:\/\/home.hirosaki-u.ac.jp\/relativity\/wp-json\/wp\/v2\/pages\/2153\/revisions\/8429"}],"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=2153"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}