{"id":6069,"date":"2023-04-07T15:07:01","date_gmt":"2023-04-07T06:07:01","guid":{"rendered":"https:\/\/home.hirosaki-u.ac.jp\/relativity\/?page_id=6069"},"modified":"2024-03-15T14:49:45","modified_gmt":"2024-03-15T05:49:45","slug":"sympy-%e3%81%a7-1-%e5%a4%89%e6%95%b0%e9%96%a2%e6%95%b0%e3%81%ae%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\/sympy-%e3%81%a7%e7%90%86%e5%b7%a5%e7%b3%bb%e3%81%ae%e6%95%b0%e5%ad%a6b\/sympy-%e3%81%a7-1-%e5%a4%89%e6%95%b0%e9%96%a2%e6%95%b0%e3%81%ae%e7%a9%8d%e5%88%86\/","title":{"rendered":"SymPy \u3067 1 \u5909\u6570\u95a2\u6570\u306e\u7a4d\u5206"},"content":{"rendered":"

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Python \u3067\u5fae\u5206\u7a4d\u5206\u306a\u3069\u306e\u8a18\u53f7\u8a08\u7b97\u3092\u3059\u308b\u5834\u5408\u306f\uff0c\u4ee5\u4e0b\u306e\u3088\u3046\u306b\u5fc5\u8981\u306a\u30d1\u30c3\u30b1\u30fc\u30b8\u3092 import \u3057\u307e\u3059\u3002\uff08\u5f18\u5927 JupyterHub \u3067\u306f\u5fc5\u8981\u306a\u30d1\u30c3\u30b1\u30fc\u30b8\u306f\u30a4\u30f3\u30b9\u30c8\u30fc\u30eb\u6e08\u307f\u3067\u3059\u3002\uff09<\/p>\n

\u30bb\u30eb\u3092\u30af\u30ea\u30c3\u30af\u3057\u3066\uff0c\u4e0a\u306e\u30e1\u30cb\u30e5\u30fc\u304b\u3089\u300c\u25b6\ufe0e Run\u300d\u3092\u30af\u30ea\u30c3\u30af\u3059\u308b\u304b\uff0cShift \u30ad\u30fc\u3092\u62bc\u3057\u306a\u304c\u3089 Enter \u30ad\u30fc\uff08Return \u30ad\u30fc\uff09\u3092\u62bc\u3059\u3068\u5b9f\u884c\u3057\u307e\u3059\u3002<\/p>\n<\/div>\n<\/div>\n<\/div>\n

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from sympy.abc<\/span> import *<\/span> \r\nfrom<\/span> sympy<\/span> import<\/span> *<\/span>\r\n\r\n# SymPy Plotting Backends (SPB): \u30b0\u30e9\u30d5\u3092\u63cf\u304f\u969b\u306b\u5229\u7528<\/span>\r\nfrom<\/span> spb<\/span> import<\/span> *<\/span>\r\n\r\n# \u30b0\u30e9\u30d5\u3092 SVG \u3067 Notebook \u306b\u30a4\u30f3\u30e9\u30a4\u30f3\u8868\u793a<\/span>\r\n%<\/span>config<\/span> InlineBackend.figure_formats = ['svg']\r\n<\/pre>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n
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\u4e0d\u5b9a\u7a4d\u5206<\/h3>\n
\u5fae\u5206\u306e\u9006\u6f14\u7b97\u3068\u3057\u3066\u306e\u4e0d\u5b9a\u7a4d\u5206\u3002
\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
\u5c0e\u95a2\u6570\u304c\u6c42\u3081\u3089\u308c\u3066\u3044\u308b\u5168\u3066\u306e\u521d\u7b49\u95a2\u6570\u306f\uff0c\u9006\u6f14\u7b97\u3068\u3057\u3066\u306e\u4e0d\u5b9a\u7a4d\u5206\u3092\u884c\u306a\u3046\u3068\u521d\u7b49\u95a2\u6570\u3092\u4f7f\u3063\u3066\u8868\u3059\u3053\u3068\u304c\u3067\u304d\u308b\u3002<\/p>\n
\u4ee5\u4e0b\u306e\u4e0d\u5b9a\u7a4d\u5206\u3067\u306f\uff0c\u7a4d\u5206\u5b9a\u6570\u306f\u7701\u7565\u3055\u308c\u3066\u51fa\u529b\u3055\u308c\u308b\u3002<\/p>\n
SymPy \u3067\u306f\u4e0d\u5b9a\u7a4d\u5206\u306f\u4ee5\u4e0b\u306e\u3088\u3046\u306b\u66f8\u304d\u307e\u3059\u3002<\/p>\n
$\\displaystyle \\int f(x) dx = $ integrate(f(x), x)<\/code><\/p>\n<\/div>\n<\/div>\n<\/div>\n
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\u3079\u304d\u95a2\u6570<\/h4>\n
$\\displaystyle \\int x^p\\, dx$ \u306e\u7a4d\u5206\u3002<\/p>\n<\/div>\n<\/div>\n<\/div>\n
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# \u3068\u3063\u3068\u3068\u7b54\u3048\u3092\u77e5\u308a\u305f\u3044\u5834\u5408\u306f...<\/span>\r\n\r\nintegrate<\/span>(<\/span>x<\/span>**<\/span>p<\/span>,<\/span> x<\/span>)<\/span>\r\n<\/pre>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n
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$\\displaystyle \\begin{cases} \\frac{x^{p + 1}}{p + 1} & \\text{for}\\: p \\neq -1 \\\\\\log{\\left(x \\right)} & \\text{otherwise} \\end{cases}$<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n
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# \u4f55\u3092\u8a08\u7b97\u3059\u308b\u304b\u3092\u8868\u793a\u3055\u305b\u3066\u304b\u3089\u5b9f\u884c\u3059\u308b\u5834\u5408\u306f...<\/span>\r\n\r\nEq<\/span>(<\/span>Integral<\/span>(<\/span>x<\/span>**<\/span>p<\/span>,<\/span> x<\/span>),<\/span> \r\n   Integral<\/span>(<\/span>x<\/span>**<\/span>p<\/span>,<\/span> x<\/span>)<\/span>.<\/span>doit<\/span>())<\/span>\r\n<\/pre>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n
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$\\displaystyle \\int x^{p}\\, dx = \\begin{cases} \\frac{x^{p + 1}}{p + 1} & \\text{for}\\: p \\neq -1 \\\\\\log{\\left(x \\right)} & \\text{otherwise} \\end{cases}$<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n
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\u6307\u6570\u95a2\u6570<\/h4>\n
$\\displaystyle \\int e^x\\, dx$ \u306e\u7a4d\u5206\u3002<\/p>\n<\/div>\n<\/div>\n<\/div>\n
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In\u00a0[4]:<\/div>\n
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Eq<\/span>(<\/span>Integral<\/span>(<\/span>exp<\/span>(<\/span>x<\/span>),<\/span> x<\/span>),<\/span> \r\n   Integral<\/span>(<\/span>exp<\/span>(<\/span>x<\/span>),<\/span> x<\/span>)<\/span>.<\/span>doit<\/span>())<\/span>\r\n<\/pre>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n
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$\\displaystyle \\int e^{x}\\, dx = e^{x}$<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n
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\u539f\u59cb\u95a2\u6570\u304c\u81ea\u7136\u5bfe\u6570\u306b\u306a\u308b\u4f8b<\/h4>\n
$\\displaystyle \\int \\frac{1}{x}\\, dx = \\log |x|$ \u3068\u306a\u308b\u306f\u305a\u3067\u3059\u304c…<\/p>\n<\/div>\n<\/div>\n<\/div>\n
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In\u00a0[5]:<\/div>\n
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Eq<\/span>(<\/span>Integral<\/span>(<\/span>1<\/span>\/<\/span>x<\/span>,<\/span> x<\/span>),<\/span> \r\n   Integral<\/span>(<\/span>1<\/span>\/<\/span>x<\/span>,<\/span> x<\/span>)<\/span>.<\/span>doit<\/span>())<\/span>\r\n<\/pre>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n
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$\\displaystyle \\int \\frac{1}{x}\\, dx = \\log{\\left(x \\right)}$<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n
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SymPy \u306e\u7a4d\u5206\u3067\u306f $\\log$ \u306e\u4e2d\u8eab\u306e\u7d76\u5bfe\u5024\u3092\u7701\u7565\u3059\u308b\u50be\u5411\u306b\u3042\u308b\u3002<\/p>\n<\/div>\n<\/div>\n<\/div>\n
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\u4e09\u89d2\u95a2\u6570\u306e\u4e0d\u5b9a\u7a4d\u5206<\/h4>\n<\/div>\n<\/div>\n<\/div>\n
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In\u00a0[6]:<\/div>\n
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Eq<\/span>(<\/span>Integral<\/span>(<\/span>cos<\/span>(<\/span>x<\/span>),<\/span> x<\/span>),<\/span> \r\n   Integral<\/span>(<\/span>cos<\/span>(<\/span>x<\/span>),<\/span> x<\/span>)<\/span>.<\/span>doit<\/span>())<\/span>\r\n<\/pre>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n
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$\\displaystyle \\int \\cos{\\left(x \\right)}\\, dx = \\sin{\\left(x \\right)}$<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n
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Eq<\/span>(<\/span>Integral<\/span>(<\/span>sin<\/span>(<\/span>x<\/span>),<\/span> x<\/span>),<\/span> \r\n   Integral<\/span>(<\/span>sin<\/span>(<\/span>x<\/span>),<\/span> x<\/span>)<\/span>.<\/span>doit<\/span>())<\/span>\r\n<\/pre>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n
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$\\displaystyle \\int \\sin{\\left(x \\right)}\\, dx = – \\cos{\\left(x \\right)}$<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n
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Eq<\/span>(<\/span>Integral<\/span>(<\/span>1<\/span>\/<\/span>cos<\/span>(<\/span>x<\/span>)<\/span>**<\/span>2<\/span>,<\/span> x<\/span>),<\/span> \r\n   Integral<\/span>(<\/span>1<\/span>\/<\/span>cos<\/span>(<\/span>x<\/span>)<\/span>**<\/span>2<\/span>,<\/span> x<\/span>)<\/span>.<\/span>doit<\/span>())<\/span>\r\n<\/pre>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n
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$\\displaystyle \\int \\frac{1}{\\cos^{2}{\\left(x \\right)}}\\, dx = \\frac{\\sin{\\left(x \\right)}}{\\cos{\\left(x \\right)}}$<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n
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# simplify \u3059\u308c\u3070 tan(x) \u306b<\/span>\r\n\r\nEq<\/span>(<\/span>integrate<\/span>(<\/span>1<\/span>\/<\/span>cos<\/span>(<\/span>x<\/span>)<\/span>**<\/span>2<\/span>,<\/span> x<\/span>),<\/span>\r\n   integrate<\/span>(<\/span>1<\/span>\/<\/span>cos<\/span>(<\/span>x<\/span>)<\/span>**<\/span>2<\/span>,<\/span> x<\/span>)<\/span>.<\/span>simplify<\/span>())<\/span>\r\n<\/pre>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n
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$\\displaystyle \\frac{\\sin{\\left(x \\right)}}{\\cos{\\left(x \\right)}} = \\tan{\\left(x \\right)}$<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n
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\u539f\u59cb\u95a2\u6570\u304c\u9006\u4e09\u89d2\u95a2\u6570\u306b\u306a\u308b\u4f8b<\/h4>\n<\/div>\n<\/div>\n<\/div>\n
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In\u00a0[10]:<\/div>\n
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Eq<\/span>(<\/span>Integral<\/span>(<\/span>1<\/span>\/<\/span>sqrt<\/span>(<\/span>1<\/span>-<\/span>x<\/span>**<\/span>2<\/span>),<\/span> x<\/span>),<\/span> \r\n   Integral<\/span>(<\/span>1<\/span>\/<\/span>sqrt<\/span>(<\/span>1<\/span>-<\/span>x<\/span>**<\/span>2<\/span>),<\/span> x<\/span>)<\/span>.<\/span>doit<\/span>())<\/span>\r\n<\/pre>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n
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$\\displaystyle \\int \\frac{1}{\\sqrt{1 – x^{2}}}\\, dx = \\operatorname{asin}{\\left(x \\right)}$<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n
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$$\\left(\\cos^{-1} x\\right)’ = – \\frac{1}{\\sqrt{1-x^2}}$$\u306a\u306e\u3067
\n$$- \\int \\frac{1}{\\sqrt{1-x^2}}\\, dx = \\cos^{-1} x$$\u3068\u306a\u308a\u305d\u3046\u3067\u3059\u304c…<\/div>\n<\/div>\n<\/div>\n
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In\u00a0[11]:<\/div>\n
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Eq<\/span>(<\/span>Integral<\/span>(<\/span>-<\/span>1<\/span>\/<\/span>sqrt<\/span>(<\/span>1<\/span>-<\/span>x<\/span>**<\/span>2<\/span>),<\/span> x<\/span>),<\/span> \r\n   Integral<\/span>(<\/span>-<\/span>1<\/span>\/<\/span>sqrt<\/span>(<\/span>1<\/span>-<\/span>x<\/span>**<\/span>2<\/span>),<\/span> x<\/span>)<\/span>.<\/span>doit<\/span>())<\/span>\r\n<\/pre>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n
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Out[11]:<\/div>\n
$\\displaystyle \\int \\left(- \\frac{1}{\\sqrt{1 – x^{2}}}\\right)\\, dx = – \\operatorname{asin}{\\left(x \\right)}$<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n
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\u5b9f\u306f
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$\\cos^{-1} x$ \u3068 $- \\sin^{-1} x$ \u306e\u9055\u3044 $\\displaystyle \\frac{\\pi}{2}$ \u306f\u7a4d\u5206\u5b9a\u6570\u306e\u4e2d\u306b\u542b\u307e\u308c\u3066\u3057\u307e\u3044\u307e\u3059\u3002<\/p>\n<\/div>\n<\/div>\n<\/div>\n
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# \u4e0d\u5b9a\u7a4d\u5206\u306e\u7b54\u3048\u3092 acos(x) \u3092\u4f7f\u3063\u3066\u66f8\u304d\u76f4\u3059\u3068...<\/span>\r\n\r\nEq<\/span>(<\/span>Integral<\/span>(<\/span>-<\/span>1<\/span>\/<\/span>sqrt<\/span>(<\/span>1<\/span>-<\/span>x<\/span>**<\/span>2<\/span>),<\/span> x<\/span>),<\/span> \r\n   integrate<\/span>(<\/span>-<\/span>1<\/span>\/<\/span>sqrt<\/span>(<\/span>1<\/span>-<\/span>x<\/span>**<\/span>2<\/span>),<\/span> x<\/span>)<\/span>.<\/span>rewrite<\/span>(<\/span>acos<\/span>))<\/span>\r\n<\/pre>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n
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Out[12]:<\/div>\n
$\\displaystyle \\int \\left(- \\frac{1}{\\sqrt{1 – x^{2}}}\\right)\\, dx = \\operatorname{acos}{\\left(x \\right)} – \\frac{\\pi}{2}$<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n
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\u5b9a\u7a4d\u5206<\/h3>\n
SymPy \u3067\u306f\u5b9a\u7a4d\u5206\u306f\u4ee5\u4e0b\u306e\u3088\u3046\u306b\u66f8\u304d\u307e\u3059\u3002<\/p>\n
$\\displaystyle \\int_a^b f(x)\\, dx = $ integrate(f(x), (x, a, b))<\/code><\/p>\n
\u7f6e\u63db\u7a4d\u5206<\/h3>\n
\u7f6e\u63db\u7a4d\u5206\u306e\u9805\u3067\u4f8b\u3068\u3057\u3066\u3042\u3052\u3066\u3044\u308b\u4e0d\u5b9a\u7a4d\u5206\uff1a$\\displaystyle\\ \\ \\int \\frac{\\log x}{x} \\, dx$<\/p>\n<\/div>\n<\/div>\n<\/div>\n
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integrate<\/span>(<\/span>log<\/span>(<\/span>x<\/span>)<\/span>\/<\/span>x<\/span>,<\/span> x<\/span>)<\/span>\r\n<\/pre>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n
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$\\displaystyle \\frac{\\log{\\left(x \\right)}^{2}}{2}$<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n
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\u7c21\u5358\u306a\u5834\u5408\u3060\u3068 integrate()<\/code> \u3067\u7a4d\u5206\u3067\u304d\u3066\u3057\u307e\u3046\u3002\u3053\u308c\u3092\u6562\u3048\u3066\u7f6e\u63db\u7a4d\u5206\u3057\u3066\u3044\u308b\u306e\u304c\u4ee5\u4e0b\u306e\u4f8b\u3002<\/p>\n
\u307e\u305a\uff0c\u4e0d\u5b9a\u7a4d\u5206\u3092 int1<\/code> \u3068\u3057\u3066\u5b9a\u7fa9\u3002<\/p>\n<\/div>\n<\/div>\n<\/div>\n
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In\u00a0[14]:<\/div>\n
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int1<\/span> =<\/span> Integral<\/span>(<\/span>log<\/span>(<\/span>x<\/span>)<\/span>\/<\/span>x<\/span>,<\/span> x<\/span>)<\/span>\r\nint1<\/span>\r\n<\/pre>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n
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Out[14]:<\/div>\n
$\\displaystyle \\int \\frac{\\log{\\left(x \\right)}}{x}\\, dx$<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n
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int1<\/code> \u306e\u4e2d\u8eab\u3092 .transform(log(x), t)<\/code> \u306b\u3088\u3063\u3066 $\\log x \\Rightarrow t$ \u3068\u3044\u3046\u5909\u6570\u306b\u7f6e\u63db\u3059\u308b\u3002<\/p>\n<\/div>\n<\/div>\n<\/div>\n
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In\u00a0[15]:<\/div>\n
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int2<\/span> =<\/span> int1<\/span>.<\/span>transform<\/span>(<\/span>log<\/span>(<\/span>x<\/span>),<\/span> t<\/span>)<\/span>\r\nint2<\/span>\r\n<\/pre>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n
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$\\displaystyle \\int \\log{\\left(e^{t} \\right)}\\, dt$<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n
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\u4e0a\u5f0f\u3067\u306f\u3055\u3089\u306b $\\log \\left(e^t\\right) = t$ \u3068\u306a\u308b\u306f\u305a\u3060\u304c\uff0c\u305d\u3053\u306f\u307b\u3063\u3066\u304a\u3044\u3066 .doit()<\/code> \u3067\u5b9f\u969b\u306b\u7a4d\u5206\u3092\u5b9f\u884c\u3055\u305b\u308b\u3002<\/p>\n<\/div>\n<\/div>\n<\/div>\n
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In\u00a0[16]:<\/div>\n
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int2<\/span>.<\/span>doit<\/span>()<\/span>\r\n<\/pre>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n
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Out[16]:<\/div>\n
$\\displaystyle \\frac{t^{2}}{2}$<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n
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\u6700\u5f8c\u306b .subs(t, log(x))<\/code> \u306b\u3088\u3063\u3066 $t$ \u3092\u3082\u3068\u306e $\\log x$ \u3067\u8868\u3059\u3068…<\/p>\n<\/div>\n<\/div>\n<\/div>\n
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In\u00a0[17]:<\/div>\n
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int2<\/span>.<\/span>doit<\/span>()<\/span>.<\/span>subs<\/span>(<\/span>t<\/span>,<\/span> log<\/span>(<\/span>x<\/span>))<\/span>\r\n<\/pre>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n
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Out[17]:<\/div>\n
$\\displaystyle \\frac{\\log{\\left(x \\right)}^{2}}{2}$<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n
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\u90e8\u5206\u7a4d\u5206<\/h3>\n
\u90e8\u5206\u7a4d\u5206\u306e\u4f8b\u3068\u3057\u3066\u3042\u3052\u3066\u3044\u308b $\\displaystyle \\int \\log x \\, dx$ \u3082\uff0c\u7279\u306b\u554f\u984c\u306a\u304f\u7a4d\u5206\u3067\u304d\u3066\u3057\u307e\u3044\u307e\u3059\u3002<\/p>\n<\/div>\n<\/div>\n<\/div>\n
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In\u00a0[18]:<\/div>\n
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Eq<\/span>(<\/span>Integral<\/span>(<\/span>log<\/span>(<\/span>x<\/span>),<\/span> x<\/span>),<\/span> \r\n   Integral<\/span>(<\/span>log<\/span>(<\/span>x<\/span>),<\/span> x<\/span>)<\/span>.<\/span>doit<\/span>())<\/span>\r\n<\/pre>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n
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Out[18]:<\/div>\n
$\\displaystyle \\int \\log{\\left(x \\right)}\\, dx = x \\log{\\left(x \\right)} – x$<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n
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<\/div>\n
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\u9006\u4e09\u89d2\u95a2\u6570\u306e\u7a4d\u5206<\/h4>\n
\u554f\uff1a
\n\u9006\u4e09\u89d2\u95a2\u6570\u3084\u9006\u53cc\u66f2\u7dda\u95a2\u6570\u306e\u5fae\u5206\u306f\u3084\u3063\u305f\u3051\u3069\uff0c\u9006\u4e09\u89d2\u95a2\u6570\u3084\u9006\u53cc\u66f2\u7dda\u95a2\u6570\u306e\u7a4d\u5206\u306f\u3069\u3046\u306a\u308b\u306e\uff1f<\/p>\n
\u7b54\uff1a
\n\u90e8\u5206\u7a4d\u5206\u3057\u3066\u304f\u3060\u3055\u3044\u3002\u57fa\u672c\u7684\u306b\u9006\u4e09\u89d2\u95a2\u6570\u3084\u9006\u53cc\u66f2\u7dda\u95a2\u6570\u306e\u5fae\u5206\u304c\u308f\u304b\u308c\u3070\uff0c\u7a4d\u5206\u3082\u3067\u304d\u307e\u3059\u3002<\/p>\n<\/div>\n<\/div>\n<\/div>\n
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In\u00a0[19]:<\/div>\n
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Eq<\/span>(<\/span>Integral<\/span>(<\/span>asin<\/span>(<\/span>x<\/span>),<\/span> x<\/span>),<\/span> \r\n   Integral<\/span>(<\/span>asin<\/span>(<\/span>x<\/span>),<\/span> x<\/span>)<\/span>.<\/span>doit<\/span>())<\/span>\r\n<\/pre>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n
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Out[19]:<\/div>\n
$\\displaystyle \\int \\operatorname{asin}{\\left(x \\right)}\\, dx = x \\operatorname{asin}{\\left(x \\right)} + \\sqrt{1 – x^{2}}$<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n
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In\u00a0[20]:<\/div>\n
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Eq<\/span>(<\/span>Integral<\/span>(<\/span>acos<\/span>(<\/span>x<\/span>),<\/span> x<\/span>),<\/span> \r\n   Integral<\/span>(<\/span>acos<\/span>(<\/span>x<\/span>),<\/span> x<\/span>)<\/span>.<\/span>doit<\/span>())<\/span>\r\n<\/pre>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n
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Out[20]:<\/div>\n
$\\displaystyle \\int \\operatorname{acos}{\\left(x \\right)}\\, dx = x \\operatorname{acos}{\\left(x \\right)} – \\sqrt{1 – x^{2}}$<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n
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In\u00a0[21]:<\/div>\n
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Eq<\/span>(<\/span>Integral<\/span>(<\/span>atan<\/span>(<\/span>x<\/span>),<\/span> x<\/span>),<\/span> \r\n   Integral<\/span>(<\/span>atan<\/span>(<\/span>x<\/span>),<\/span> x<\/span>)<\/span>.<\/span>doit<\/span>())<\/span>\r\n<\/pre>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n
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Out[21]:<\/div>\n
$\\displaystyle \\int \\operatorname{atan}{\\left(x \\right)}\\, dx = x \\operatorname{atan}{\\left(x \\right)} – \\frac{\\log{\\left(x^{2} + 1 \\right)}}{2}$<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n
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<\/div>\n
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\u9006\u53cc\u66f2\u7dda\u95a2\u6570\u306e\u7a4d\u5206<\/h4>\n<\/div>\n<\/div>\n<\/div>\n
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In\u00a0[22]:<\/div>\n
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Eq<\/span>(<\/span>Integral<\/span>(<\/span>asinh<\/span>(<\/span>x<\/span>),<\/span> x<\/span>),<\/span> \r\n   Integral<\/span>(<\/span>asinh<\/span>(<\/span>x<\/span>),<\/span> x<\/span>)<\/span>.<\/span>doit<\/span>())<\/span>\r\n<\/pre>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n
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Out[22]:<\/div>\n
$\\displaystyle \\int \\operatorname{asinh}{\\left(x \\right)}\\, dx = x \\operatorname{asinh}{\\left(x \\right)} – \\sqrt{x^{2} + 1}$<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n
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\u672c\u7a3f\u57f7\u7b46\u6642\u70b9\u3067\u306f\uff0cSymPy \u3067\u306f\u306a\u305c\u304b $\\cosh^{-1} x$ \u306e\u7a4d\u5206\u304c\u3067\u304d\u306a\u3044\u3002\u30d0\u30b0\uff1f<\/p>\n<\/div>\n<\/div>\n<\/div>\n
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In\u00a0[23]:<\/div>\n
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integrate<\/span>(<\/span>acosh<\/span>(<\/span>x<\/span>),<\/span> x<\/span>)<\/span>\r\n<\/pre>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n
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Out[23]:<\/div>\n
$\\displaystyle \\int \\operatorname{acosh}{\\left(x \\right)}\\, dx$<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n
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\u3057\u304b\u305f\u304c\u306a\u3044\u306e\u3067\uff0c\u307e\u305a\u306f\u4eba\u529b\u3067\u90e8\u5206\u7a4d\u5206\u306e\u5f62\u306b\u3057\u307e\u3059\u3002<\/p>\n
\\begin{eqnarray}
\n\\int \\operatorname{acosh}{x}\\, dx &=&
\nx \\operatorname{acosh}{x} – \\int x \\frac{d}{dx} \\operatorname{acosh}{x} \\,dx
\n\\end{eqnarray}<\/p>\n
$\\operatorname{acosh}{x}$ \u306e\u5fae\u5206\u306f\uff0cSymPy \u3067\u306f…<\/p>\n<\/div>\n<\/div>\n<\/div>\n
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In\u00a0[24]:<\/div>\n
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diff<\/span>(<\/span>acosh<\/span>(<\/span>x<\/span>),<\/span> x<\/span>)<\/span>\r\n<\/pre>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n
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Out[24]:<\/div>\n
$\\displaystyle \\frac{1}{\\sqrt{x – 1} \\sqrt{x + 1}}$<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n
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SymPy \u306f\u3069\u3046\u3082\u3053\u306e\u624b\u306e\u7a4d\u5206\u304c\u82e6\u624b\u306e\u3088\u3046\u3067\u3059\u3002<\/p>\n<\/div>\n<\/div>\n<\/div>\n
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In\u00a0[25]:<\/div>\n
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Integral<\/span>(<\/span>x<\/span>\/<\/span>(<\/span>sqrt<\/span>(<\/span>x<\/span>-<\/span>1<\/span>)<\/span>*<\/span>sqrt<\/span>(<\/span>x<\/span>+<\/span>1<\/span>)),<\/span> x<\/span>)<\/span>\r\n<\/pre>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n
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Out[25]:<\/div>\n
$\\displaystyle \\int \\frac{x}{\\sqrt{x – 1} \\sqrt{x + 1}}\\, dx$<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n
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\u5206\u6bcd\u3092\u4ee5\u4e0b\u306e\u3088\u3046\u306b\u66f8\u304d\u63db\u3048\u308b\u3068\u7c21\u5358\u306b\u7b54\u3048\u3092\u3060\u3057\u307e\u3059\u3002<\/p>\n<\/div>\n<\/div>\n<\/div>\n
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In\u00a0[26]:<\/div>\n
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Eq<\/span>(<\/span>Integral<\/span>(<\/span>x<\/span>\/<\/span>(<\/span>sqrt<\/span>(<\/span>x<\/span>**<\/span>2<\/span>-<\/span>1<\/span>)),<\/span> x<\/span>),<\/span> \r\n   Integral<\/span>(<\/span>x<\/span>\/<\/span>(<\/span>sqrt<\/span>(<\/span>x<\/span>**<\/span>2<\/span>-<\/span>1<\/span>)),<\/span> x<\/span>)<\/span>.<\/span>doit<\/span>())<\/span>\r\n<\/pre>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n
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Out[26]:<\/div>\n
$\\displaystyle \\int \\frac{x}{\\sqrt{x^{2} – 1}}\\, dx = \\sqrt{x^{2} – 1}$<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n
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\u3068\u3044\u3046\u3053\u3068\u3067\uff0c\u7d50\u5c40…<\/p>\n<\/div>\n<\/div>\n<\/div>\n
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In\u00a0[27]:<\/div>\n
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Eq<\/span>(<\/span>Integral<\/span>(<\/span>acosh<\/span>(<\/span>x<\/span>),<\/span> x<\/span>),<\/span> \r\n   x<\/span> *<\/span> acosh<\/span>(<\/span>x<\/span>)<\/span> -<\/span> integrate<\/span>(<\/span>x<\/span>\/<\/span>(<\/span>sqrt<\/span>(<\/span>x<\/span>**<\/span>2<\/span>-<\/span>1<\/span>)),<\/span> x<\/span>))<\/span>\r\n<\/pre>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n
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Out[27]:<\/div>\n
$\\displaystyle \\int \\operatorname{acosh}{\\left(x \\right)}\\, dx = x \\operatorname{acosh}{\\left(x \\right)} – \\sqrt{x^{2} – 1}$<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n
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In\u00a0[28]:<\/div>\n
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Eq<\/span>(<\/span>Integral<\/span>(<\/span>atanh<\/span>(<\/span>x<\/span>),<\/span> x<\/span>),<\/span> \r\n   integrate<\/span>(<\/span>atanh<\/span>(<\/span>x<\/span>),<\/span> x<\/span>))<\/span>\r\n<\/pre>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n
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Out[28]:<\/div>\n
$\\displaystyle \\int \\operatorname{atanh}{\\left(x \\right)}\\, dx = x \\operatorname{atanh}{\\left(x \\right)} + \\log{\\left(x + 1 \\right)} – \\operatorname{atanh}{\\left(x \\right)}$<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n
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\u3053\u308c\u3067\u3082\u3044\u3044\u3067\u3059\u304c\uff0c$\\displaystyle\\int \\operatorname{atan}{\\left(x \\right)}\\, dx$ \u3068\u306e\u5bfe\u5fdc\u304c\u308f\u304b\u308a\u3084\u3059\u3044\u5f62\u306b\u3057\u3066\u307f\u307e\u3059\u3002<\/p>\n
\u307e\u305a\u306f\u4eba\u529b\u3067\u90e8\u5206\u7a4d\u5206\u306e\u5f62\u306b\u3057\u3066…<\/p>\n
\\begin{eqnarray}
\n\\int \\operatorname{atanh}{\\left(x \\right)}\\, dx &=&
\nx \\operatorname{atanh}{\\left(x \\right)} – \\int x \\frac{d}{dx}\\operatorname{atanh}{\\left(x \\right)}\\,dx
\n\\end{eqnarray}<\/p>\n
$\\operatorname{atanh}{\\left(x \\right)}$ \u306e\u5fae\u5206\u306f SymPy \u3067\u306f…<\/p>\n<\/div>\n<\/div>\n<\/div>\n
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In\u00a0[29]:<\/div>\n
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diff<\/span>(<\/span>atanh<\/span>(<\/span>x<\/span>),<\/span> x<\/span>)<\/span>\r\n<\/pre>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n
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Out[29]:<\/div>\n
$\\displaystyle \\frac{1}{1 – x^{2}}$<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n
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In\u00a0[30]:<\/div>\n
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Eq<\/span>(<\/span>Integral<\/span>(<\/span>atanh<\/span>(<\/span>x<\/span>),<\/span> x<\/span>),<\/span> \r\n   x<\/span>*<\/span> atanh<\/span>(<\/span>x<\/span>)<\/span> -<\/span> integrate<\/span>(<\/span>x<\/span>\/<\/span>(<\/span>1<\/span>-<\/span>x<\/span>**<\/span>2<\/span>),<\/span> x<\/span>))<\/span>\r\n<\/pre>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n
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Out[30]:<\/div>\n
$\\displaystyle \\int \\operatorname{atanh}{\\left(x \\right)}\\, dx = x \\operatorname{atanh}{\\left(x \\right)} + \\frac{\\log{\\left(x^{2} – 1 \\right)}}{2}$<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n
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\u60dc\u3057\u3044\uff01 $\\operatorname{atanh}{\\left(x \\right)}$ \u306e $x$ \u306e\u3068\u308a\u3046\u308b\u7bc4\u56f2\u306f $-1 < x < 1$ \u306a\u306e\u3067\uff0c\u6700\u5f8c\u306e $\\log$ \u306e\u9805\u306f<\/p>\n
$\\displaystyle \\int \\operatorname{atanh}{\\left(x \\right)}\\, dx = x \\operatorname{atanh}{\\left(x \\right)} + \\frac{\\log{\\left(1-x^{2} \\right)}}{2}$<\/p>\n
\u3068\u66f8\u304f\u3079\u304d\u3060\u3088\u306d\u3047\u3002<\/p>\n<\/div>\n<\/div>\n<\/div>\n
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\u6709\u7406\u95a2\u6570\u306e\u7a4d\u5206<\/h3>\n
$$f(x) = \\frac{2 x^3 + 3 x^2 \u2013 2 x \u2013 1}{x^2 + x \u2013 2}$$<\/p>\n
\u306e\u3088\u3046\u306b $\\displaystyle \\frac{\\mbox{\u591a\u9805\u5f0f}}{\\mbox{\u591a\u9805\u5f0f}}$ \u306e\u5f62\u306b\u306a\u3063\u3066\u3044\u308b\u95a2\u6570\u3092\u6709\u7406\u95a2\u6570<\/strong>\u3068\u3044\u3046\u3002<\/p>\n
\u6709\u7406\u95a2\u6570\u3092\u7a4d\u5206\u3059\u308b\u969b\u306f\uff0c\u90e8\u5206\u5206\u6570\u306b\u5206\u89e3\u3057\u3066\u304b\u3089\u7a4d\u5206\u3059\u308b\u3002<\/p>\n
SymPy \u3067\u306f\u7279\u306b\u6c17\u306b\u305b\u305a integrate()<\/code> \u3067\u7a4d\u5206\u3067\u304d\u3066\u3057\u307e\u3046\u3002<\/p>\n<\/div>\n<\/div>\n<\/div>\n
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In\u00a0[31]:<\/div>\n
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f<\/span> =<\/span> (<\/span>2<\/span> *<\/span> x<\/span>**<\/span>3<\/span> +<\/span> 3<\/span> *<\/span> x<\/span>**<\/span>2<\/span> -<\/span> 2<\/span> *<\/span> x<\/span> -<\/span> 1<\/span>)<\/span>\/<\/span>(<\/span>x<\/span>**<\/span>2<\/span> +<\/span> x<\/span> -<\/span> 2<\/span>)<\/span>\r\nf<\/span>\r\n<\/pre>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n
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Out[31]:<\/div>\n
$\\displaystyle \\frac{2 x^{3} + 3 x^{2} – 2 x – 1}{x^{2} + x – 2}$<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n
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In\u00a0[32]:<\/div>\n
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integrate<\/span>(<\/span>f<\/span>,<\/span> x<\/span>)<\/span>\r\n<\/pre>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n
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Out[32]:<\/div>\n
$\\displaystyle x^{2} + x + \\frac{2 \\log{\\left(x – 1 \\right)}}{3} + \\frac{\\log{\\left(x + 2 \\right)}}{3}$<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n
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\u90e8\u5206\u5206\u6570\u306b\u5206\u89e3<\/h4>\n
\u4e0a\u8a18\u306e\u3088\u3046\u306b\uff0cSymPy \u3067\u306f\u7279\u306b\u90e8\u5206\u5206\u6570\u306b\u5206\u89e3\u3057\u306a\u304f\u3066\u3082\u7a4d\u5206\u3066\u304d\u3067\u3057\u307e\u3046\u306e\u3067\u3042\u308b\u304c\uff0c\u305d\u3053\u3092\u3042\u3048\u3066 apart()<\/code> \u95a2\u6570\u3067\u90e8\u5206\u5206\u6570\u306b\u5206\u89e3\u3057\u3066\u307f\u308b\u3002<\/p>\n<\/div>\n<\/div>\n<\/div>\n
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In\u00a0[33]:<\/div>\n
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af<\/span> =<\/span> apart<\/span>(<\/span>f<\/span>)<\/span>\r\naf<\/span>\r\n<\/pre>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n
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Out[33]:<\/div>\n
$\\displaystyle 2 x + 1 + \\frac{1}{3 \\left(x + 2\\right)} + \\frac{2}{3 \\left(x – 1\\right)}$<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n
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\u90e8\u5206\u5206\u6570\u306b\u5206\u89e3\u3059\u308b\u3053\u3068\u3067\uff0c\u4ee5\u4e0b\u306e\u3088\u3046\u306a\u7a4d\u5206\u3092\u3059\u308c\u3070\u3044\u3044\u306e\u3060\u306a\u3041\u3068\u7406\u89e3\u3067\u304d\u308b\u3002<\/p>\n
\\begin{eqnarray}
\n\\int f(x) dx &=&
\n\\int (2x+1) dx + \\frac{1}{3} \\int \\frac{dx}{x+2} + \\frac{2}{3}\\int \\frac{dx}{x-1} \\\\
\n&=& x^2 + x + \\frac{1}{3} \\log(x+2) + \\frac{2}{3} \\log(x-1)
\n\\end{eqnarray}<\/p>\n<\/div>\n<\/div>\n<\/div>\n
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In\u00a0[34]:<\/div>\n
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Eq<\/span>(<\/span>Integral<\/span>(<\/span>af<\/span>,<\/span> x<\/span>),<\/span> \r\n   Integral<\/span>(<\/span>af<\/span>,<\/span> x<\/span>)<\/span>.<\/span>doit<\/span>())<\/span>\r\n<\/pre>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n
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Out[34]:<\/div>\n
$\\displaystyle \\int \\left(2 x + 1 + \\frac{1}{3 \\left(x + 2\\right)} + \\frac{2}{3 \\left(x – 1\\right)}\\right)\\, dx = x^{2} + x + \\frac{2 \\log{\\left(x – 1 \\right)}}{3} + \\frac{\\log{\\left(x + 2 \\right)}}{3}$<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n
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$\\sin x, \\cos x$ \u306e\u6709\u7406\u95a2\u6570\u306e\u7a4d\u5206<\/h3>\n
\u305f\u3068\u3048\u3070\uff0c$\\displaystyle \\frac{(\\sin x)^2}{1 + \\cos x + 2 \\sin x}$ \u306e\u3088\u3046\u306a\uff0c $\\sin x$ \u3068 $\\cos x$ \u306e\u6709\u7406\u95a2\u6570\u306e\u5f62\u306e\u95a2\u6570\u306e\u7a4d\u5206\u3002<\/p>\n
\u6559\u79d1\u66f8\u7684\u306b\u306f $\\displaystyle \\tan \\frac{x}{2} \\equiv t$ \u3068\u3044\u3046\u5909\u6570\u5909\u63db\u3092\u3057\u3066\u7f6e\u63db\u7a4d\u5206\u3059\u308c\u3070\u3088\u3044\u3068\u3044\u3046\u3053\u3068\u306b\u306a\u3063\u3066\u3044\u308b\u3002<\/p>\n<\/div>\n<\/div>\n<\/div>\n
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\u7df4\u7fd2\u554f\u984c 1<\/h4>\n
$\\displaystyle \\int \\frac{1}{\\cos x} \\,dx$<\/p>\n
\u3053\u308c\u306f\u305d\u306e\u307e\u307e\u306e\u5f62\u3067 integrate()<\/code> \u3067\u304d\u308b\u3002<\/p>\n<\/div>\n<\/div>\n<\/div>\n
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In\u00a0[35]:<\/div>\n
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ans1<\/span> =<\/span> integrate<\/span>(<\/span>1<\/span>\/<\/span>cos<\/span>(<\/span>x<\/span>),<\/span> x<\/span>)<\/span>\r\nEq<\/span>(<\/span>Integral<\/span>(<\/span>1<\/span>\/<\/span>cos<\/span>(<\/span>x<\/span>),<\/span> x<\/span>),<\/span> ans1<\/span>)<\/span>\r\n<\/pre>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n
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Out[35]:<\/div>\n
$\\displaystyle \\int \\frac{1}{\\cos{\\left(x \\right)}}\\, dx = – \\frac{\\log{\\left(\\sin{\\left(x \\right)} – 1 \\right)}}{2} + \\frac{\\log{\\left(\\sin{\\left(x \\right)} + 1 \\right)}}{2}$<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n
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\u6559\u80b2\u7684\u898b\u5730\u304b\u3089\u7f6e\u63db\u7a4d\u5206\u3067\u3082\u3084\u3063\u3066\u307f\u308b\u3002<\/p>\n
$\\displaystyle \\tan \\frac{x}{2} \\equiv t$ \u3068\u304a\u3044\u3066…<\/p>\n<\/div>\n<\/div>\n<\/div>\n
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In\u00a0[36]:<\/div>\n
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int1<\/span> =<\/span> Integral<\/span>(<\/span>1<\/span>\/<\/span>cos<\/span>(<\/span>x<\/span>),<\/span> x<\/span>)<\/span>\r\nint2<\/span> =<\/span> int1<\/span>.<\/span>transform<\/span>(<\/span>tan<\/span>(<\/span>x<\/span>\/<\/span>2<\/span>),<\/span> t<\/span>)<\/span>.<\/span>trigsimp<\/span>()<\/span>\r\nint2<\/span>\r\n<\/pre>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n
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Out[36]:<\/div>\n
$\\displaystyle \\int \\frac{2}{1 – t^{2}}\\, dt$<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n
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\u3053\u308c\u306f\uff08\u88ab\u7a4d\u5206\u95a2\u6570\u3092\u90e8\u5206\u5206\u6570\u306b\u5206\u89e3\u3059\u308c\u3070\uff09\u7c21\u5358\u306b\u7a4d\u5206\u3067\u304d\u3066…<\/p>\n<\/div>\n<\/div>\n<\/div>\n
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In\u00a0[37]:<\/div>\n
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ans<\/span> =<\/span> int2<\/span>.<\/span>doit<\/span>()<\/span>\r\nans<\/span>\r\n<\/pre>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n
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Out[37]:<\/div>\n
$\\displaystyle – \\log{\\left(t – 1 \\right)} + \\log{\\left(t + 1 \\right)}$<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n
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\u6700\u5f8c\u306b $t$ \u3092 $\\displaystyle \\tan \\frac{x}{2}$ \u306b\u623b\u3057\u3066…<\/p>\n<\/div>\n<\/div>\n<\/div>\n
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In\u00a0[38]:<\/div>\n
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ans2<\/span> =<\/span> ans<\/span>.<\/span>subs<\/span>(<\/span>t<\/span>,<\/span> tan<\/span>(<\/span>x<\/span>\/<\/span>2<\/span>))<\/span>\r\nans2<\/span>\r\n<\/pre>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n
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Out[38]:<\/div>\n
$\\displaystyle – \\log{\\left(\\tan{\\left(\\frac{x}{2} \\right)} – 1 \\right)} + \\log{\\left(\\tan{\\left(\\frac{x}{2} \\right)} + 1 \\right)}$<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n
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\u3068\u3044\u3046\u308f\u3051\u3067\uff0c\u4e00\u898b\u8868\u793a\u306e\u7570\u306a\u308b\u539f\u59cb\u95a2\u6570\u304c\u5f97\u3089\u308c\u305f\u3002\uff08\u7d76\u5bfe\u5024\u3092\u7701\u7565\u3057\u3066\u3044\u308b\u3051\u3069\u3002\uff09<\/p>\n
\\begin{eqnarray}
\n\\int \\frac{1}{\\cos{\\left(x \\right)}}\\, dx &=&
\n\\frac{1}{2}\\log\\frac{1+\\sin x}{1-\\sin x} \\\\
\n&=& \\log\\left|\\frac{1+\\tan \\frac{x}{2}}{1-\\tan \\frac{x}{2}} \\right|
\n\\end{eqnarray}<\/p>\n
\u3053\u308c\u3089\u304c\u540c\u7b49\u3067\u3042\u308b\u3053\u3068\u3092\u793a\u3057\u3066\u304a\u3044\u3066\u304f\u3060\u3055\u3044\u3002<\/p>\n
\u30d2\u30f3\u30c8\uff1a\u305f\u3068\u3048\u3070 $ \\sin x = 2 \\sin\\frac{x}{2} \\cos\\frac{x}{2}$ \u3092\u4f7f\u3046\u3068\u304b\u3002<\/p>\n<\/div>\n<\/div>\n<\/div>\n
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\u7df4\u7fd2\u554f\u984c 2<\/h4>\n
$\\displaystyle\\int \\frac{a \u2013 b \\cos\\phi}{a^2 + b^2 -2 a b \\cos \\phi} d\\phi$<\/p>\n<\/div>\n<\/div>\n<\/div>\n
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In\u00a0[39]:<\/div>\n
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a<\/span> =<\/span> Symbol<\/span>(<\/span>'a'<\/span>,<\/span> positive<\/span> =<\/span> True<\/span>)<\/span>\r\nb<\/span> =<\/span> Symbol<\/span>(<\/span>'b'<\/span>,<\/span> positive<\/span> =<\/span> True<\/span>)<\/span>\r\nvar<\/span>(<\/span>'phi'<\/span>)<\/span>\r\n\r\nint1<\/span> =<\/span> Integral<\/span>((<\/span>a<\/span>-<\/span>b<\/span>*<\/span>cos<\/span>(<\/span>phi<\/span>))<\/span>\/<\/span>(<\/span>a<\/span>**<\/span>2<\/span> +<\/span> b<\/span>**<\/span>2<\/span> -<\/span> 2<\/span>*<\/span>a<\/span>*<\/span>b<\/span>*<\/span>cos<\/span>(<\/span>phi<\/span>)),<\/span> phi<\/span>)<\/span>\r\nint1<\/span>\r\n<\/pre>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n
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Out[39]:<\/div>\n
$\\displaystyle \\int \\frac{a – b \\cos{\\left(\\phi \\right)}}{a^{2} – 2 a b \\cos{\\left(\\phi \\right)} + b^{2}}\\, d\\phi$<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n
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In\u00a0[40]:<\/div>\n
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int1<\/span>.<\/span>doit<\/span>()<\/span>\r\n<\/pre>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n
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Out[40]:<\/div>\n
$\\displaystyle \\frac{\\phi}{2 a} – \\frac{i \\log{\\left(- \\frac{i a}{a + b} + \\frac{i b}{a + b} + \\tan{\\left(\\frac{\\phi}{2} \\right)} \\right)}}{2 a} + \\frac{i \\log{\\left(\\frac{i a}{a + b} – \\frac{i b}{a + b} + \\tan{\\left(\\frac{\\phi}{2} \\right)} \\right)}}{2 a}$<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n
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\u3044\u3061\u304a\u3046\u7a4d\u5206\u306f\u3067\u304d\u3066\u3044\u308b\u3088\u3046\u3060\u304c\uff0c\u865a\u6570\u5358\u4f4d $i$ \u304c\u73fe\u308c\u3066\u3044\u3066\u610f\u5473\u4e0d\u660e\u3002<\/p>\n
\u30bb\u30aa\u30ea\u30fc\u306b\u5f93\u3063\u3066 $\\displaystyle \\tan \\frac{\\phi}{2} \\equiv t$ \u3068\u304a\u3044\u3066…<\/p>\n<\/div>\n<\/div>\n<\/div>\n
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In\u00a0[41]:<\/div>\n
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int2<\/span> =<\/span> int1<\/span>.<\/span>transform<\/span>(<\/span>tan<\/span>(<\/span>phi<\/span>\/<\/span>2<\/span>),<\/span> t<\/span>)<\/span>.<\/span>trigsimp<\/span>()<\/span>\r\nint2<\/span>\r\n<\/pre>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n
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Out[41]:<\/div>\n
$\\displaystyle \\int \\frac{2 \\left(a – \\frac{b \\left(1 – t^{2}\\right)}{t^{2} + 1}\\right)}{\\left(t^{2} + 1\\right) \\left(a^{2} – \\frac{2 a b \\left(1 – t^{2}\\right)}{t^{2} + 1} + b^{2}\\right)}\\, dt$<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n
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\u88ab\u7a4d\u5206\u95a2\u6570\u3092\u90e8\u5206\u5206\u6570\u306b\u5206\u89e3\u3057\u3066\u307f\u308b\u3002<\/p>\n<\/div>\n<\/div>\n<\/div>\n
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In\u00a0[42]:<\/div>\n
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ft<\/span> =<\/span> diff<\/span>(<\/span>int2<\/span>,<\/span> t<\/span>)<\/span>.<\/span>apart<\/span>(<\/span>t<\/span>)<\/span>\r\nft<\/span>\r\n<\/pre>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n
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Out[42]:<\/div>\n
$\\displaystyle \\frac{\\left(a – b\\right) \\left(a + b\\right)}{a \\left(a^{2} t^{2} + a^{2} + 2 a b t^{2} – 2 a b + b^{2} t^{2} + b^{2}\\right)} + \\frac{1}{a \\left(t^{2} + 1\\right)}$<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n
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\u305d\u308c\u305e\u308c\u306e\u9805\u3092\u7a4d\u5206\u3057\u3066\u307f\u308b\u3002<\/p>\n<\/div>\n<\/div>\n<\/div>\n
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In\u00a0[43]:<\/div>\n
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term1<\/span> =<\/span> 1<\/span>\/<\/span>a<\/span>\/<\/span>(<\/span>t<\/span>**<\/span>2<\/span>+<\/span>1<\/span>)<\/span>\r\nEq<\/span>(<\/span>Integral<\/span>(<\/span>term1<\/span>,<\/span> t<\/span>)<\/span> +<\/span> Integral<\/span>((<\/span>ft<\/span>-<\/span>term1<\/span>),<\/span> t<\/span>),<\/span> \r\n   integrate<\/span>(<\/span>term1<\/span>,<\/span> t<\/span>)<\/span> +<\/span>integrate<\/span>((<\/span>ft<\/span>-<\/span>term1<\/span>),<\/span> t<\/span>)<\/span>.<\/span>simplify<\/span>())<\/span>\r\n<\/pre>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n
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Out[43]:<\/div>\n
$\\displaystyle \\int \\frac{1}{a \\left(t^{2} + 1\\right)}\\, dt + \\int \\frac{\\left(a – b\\right) \\left(a + b\\right)}{a \\left(a^{2} t^{2} + a^{2} + 2 a b t^{2} – 2 a b + b^{2} t^{2} + b^{2}\\right)}\\, dt = \\frac{\\operatorname{atan}{\\left(t \\right)}}{a} + \\frac{\\operatorname{atan}{\\left(\\frac{t \\left(a + b\\right)}{a – b} \\right)}}{a}$<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n
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\u304b\u306a\u308a\u3059\u3063\u304d\u308a\u3057\u3066\u304d\u305f\u3002\u6700\u5f8c\u306b $t$ \u3092 $\\displaystyle \\tan \\frac{\\phi}{2}$ \u306b\u623b\u3057\u3066…<\/p>\n<\/div>\n<\/div>\n<\/div>\n
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In\u00a0[44]:<\/div>\n
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ans<\/span> =<\/span> integrate<\/span>(<\/span>term1<\/span>,<\/span> t<\/span>)<\/span> +<\/span>integrate<\/span>((<\/span>ft<\/span>-<\/span>term1<\/span>),<\/span> t<\/span>)<\/span>.<\/span>simplify<\/span>()<\/span>\r\nans<\/span>.<\/span>subs<\/span>(<\/span>t<\/span>,<\/span> tan<\/span>(<\/span>phi<\/span>\/<\/span>2<\/span>))<\/span>\r\n<\/pre>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n
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Out[44]:<\/div>\n
$\\displaystyle \\frac{\\operatorname{atan}{\\left(\\frac{\\left(a + b\\right) \\tan{\\left(\\frac{\\phi}{2} \\right)}}{a – b} \\right)}}{a} + \\frac{\\operatorname{atan}{\\left(\\tan{\\left(\\frac{\\phi}{2} \\right)} \\right)}}{a}$<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n
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\u7df4\u7fd2\u554f\u984c 3<\/h4>\n
$\\displaystyle\\int \\frac{a \u2013 b \\cos\\theta}{\\left(a^2 + b^2 -2 a b \\cos \\theta\\right)^{\\frac{3}{2}}} \\sin\\theta\\, d\\theta$<\/p>\n<\/div>\n<\/div>\n<\/div>\n
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In\u00a0[45]:<\/div>\n
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var<\/span>(<\/span>'theta'<\/span>)<\/span>\r\nint1<\/span> =<\/span> Integral<\/span>(<\/span>\r\n    (<\/span>a<\/span>-<\/span>b<\/span>*<\/span>cos<\/span>(<\/span>theta<\/span>))<\/span>\/<\/span>(<\/span>sqrt<\/span>(<\/span>a<\/span>**<\/span>2<\/span>+<\/span>b<\/span>**<\/span>2<\/span>-<\/span>2<\/span>*<\/span>a<\/span>*<\/span>b<\/span>*<\/span>cos<\/span>(<\/span>theta<\/span>)))<\/span>**<\/span>3<\/span>*<\/span>sin<\/span>(<\/span>theta<\/span>),<\/span> theta<\/span>)<\/span>\r\nint1<\/span>\r\n<\/pre>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n
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Out[45]:<\/div>\n
$\\displaystyle \\int \\frac{\\left(a – b \\cos{\\left(\\theta \\right)}\\right) \\sin{\\left(\\theta \\right)}}{\\left(a^{2} – 2 a b \\cos{\\left(\\theta \\right)} + b^{2}\\right)^{\\frac{3}{2}}}\\, d\\theta$<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n
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integrate()<\/code> \u3067\u7a4d\u5206\u3067\u304d\u307e\u3059\u3002<\/p>\n<\/div>\n<\/div>\n<\/div>\n
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In\u00a0[46]:<\/div>\n
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int1<\/span>.<\/span>doit<\/span>()<\/span>.<\/span>factor<\/span>()<\/span>\r\n<\/pre>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n
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Out[46]:<\/div>\n
$\\displaystyle – \\frac{a \\cos{\\left(\\theta \\right)} – b}{a^{2} \\sqrt{a^{2} – 2 a b \\cos{\\left(\\theta \\right)} + b^{2}}}$<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n
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\u3042\u3048\u3066\u7f6e\u63db\u7a4d\u5206\u3067\u3084\u3063\u3066\u307f\u308b\u3068\uff0c$u \\equiv \\cos\\theta$ \u3068\u304a\u3044\u3066…<\/p>\n<\/div>\n<\/div>\n<\/div>\n
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\u7121\u7406\u95a2\u6570\u306e\u7a4d\u5206<\/h3>\n<\/div>\n<\/div>\n<\/div>\n
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\u4f8b 1<\/h4>\n
$\\displaystyle \\int \\frac{dx}{x \\sqrt{x+1}}$<\/p>\n
SymPy \u3067\u306f\u305d\u306e\u307e\u307e integrate()<\/code> \u3067\u304d\u307e\u3059\u3002<\/p>\n<\/div>\n<\/div>\n<\/div>\n
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In\u00a0[47]:<\/div>\n
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integrate<\/span>(<\/span>1<\/span>\/<\/span>(<\/span>x<\/span>*<\/span>sqrt<\/span>(<\/span>x<\/span>+<\/span>1<\/span>)),<\/span> x<\/span>)<\/span>\r\n<\/pre>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n
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Out[47]:<\/div>\n
$\\displaystyle \\begin{cases} – 2 \\operatorname{acoth}{\\left(\\sqrt{x + 1} \\right)} & \\text{for}\\: \\left|{x + 1}\\right| > 1 \\\\- 2 \\operatorname{atanh}{\\left(\\sqrt{x + 1} \\right)} & \\text{otherwise} \\end{cases}$<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n
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$\\log$ \u3067\u3042\u3089\u308f\u3059\u3068…<\/p>\n<\/div>\n<\/div>\n<\/div>\n
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In\u00a0[48]:<\/div>\n
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integrate<\/span>(<\/span>1<\/span>\/<\/span>(<\/span>x<\/span>*<\/span>sqrt<\/span>(<\/span>x<\/span>+<\/span>1<\/span>)),<\/span> x<\/span>)<\/span>.<\/span>rewrite<\/span>(<\/span>log<\/span>)<\/span>\r\n<\/pre>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n
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Out[48]:<\/div>\n
$\\displaystyle \\begin{cases} \\log{\\left(1 – \\frac{1}{\\sqrt{x + 1}} \\right)} – \\log{\\left(1 + \\frac{1}{\\sqrt{x + 1}} \\right)} & \\text{for}\\: \\left|{x + 1}\\right| > 1 \\\\\\log{\\left(1 – \\sqrt{x + 1} \\right)} – \\log{\\left(\\sqrt{x + 1} + 1 \\right)} & \\text{otherwise} \\end{cases}$<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n
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\u3053\u308c\u3092\u3042\u3048\u3066 $\\sqrt{x+1} = t$ \u3068\u304a\u3044\u3066\u7f6e\u63db\u7a4d\u5206\u3057\u3066\u307f\u308b\u3068…<\/p>\n<\/div>\n<\/div>\n<\/div>\n
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In\u00a0[49]:<\/div>\n
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int1<\/span> =<\/span> Integral<\/span>(<\/span>1<\/span>\/<\/span>(<\/span>x<\/span>*<\/span>sqrt<\/span>(<\/span>x<\/span>+<\/span>1<\/span>)),<\/span> x<\/span>)<\/span>\r\nint1<\/span>\r\n<\/pre>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n
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Out[49]:<\/div>\n
$\\displaystyle \\int \\frac{1}{x \\sqrt{x + 1}}\\, dx$<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n
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In\u00a0[50]:<\/div>\n
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int2<\/span> =<\/span> int1<\/span>.<\/span>transform<\/span>(<\/span>sqrt<\/span>(<\/span>x<\/span>+<\/span>1<\/span>),<\/span> t<\/span>)<\/span>\r\nint2<\/span>\r\n<\/pre>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n
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Out[50]:<\/div>\n
$\\displaystyle \\int \\frac{2 t}{\\left(t^{2} – 1\\right) \\sqrt{t^{2}}}\\, dt$<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n
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In\u00a0[51]:<\/div>\n
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ans<\/span> =<\/span> int2<\/span>.<\/span>doit<\/span>()<\/span>\r\nans<\/span>\r\n<\/pre>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n
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Out[51]:<\/div>\n
$\\displaystyle \\log{\\left(\\sqrt{t^{2}} – 1 \\right)} – \\log{\\left(\\sqrt{t^{2}} + 1 \\right)}$<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n
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<\/div>\n
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\n
\u6700\u5f8c\u306b $t$ \u3092\u3082\u3068\u306e $\\sqrt{x+1}$ \u306b\u623b\u3057\u3066\u3084\u3063\u3066…<\/p>\n<\/div>\n<\/div>\n<\/div>\n
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In\u00a0[52]:<\/div>\n
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ans<\/span>.<\/span>subs<\/span>(<\/span>t<\/span>,<\/span> sqrt<\/span>(<\/span>x<\/span>+<\/span>1<\/span>))<\/span>\r\n<\/pre>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n
\n
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Out[52]:<\/div>\n
$\\displaystyle \\log{\\left(\\sqrt{x + 1} – 1 \\right)} – \\log{\\left(\\sqrt{x + 1} + 1 \\right)}$<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n
\n
<\/div>\n
\n
\n
SymPy \u306f\u3042\u3044\u304b\u3089\u308f\u305a $\\log$ \u306e\u4e2d\u8eab\u306e\u7d76\u5bfe\u5024\u3092\u7701\u7565\u3057\u3066\u3044\u307e\u3059\u3002<\/p>\n<\/div>\n<\/div>\n<\/div>\n
\n
<\/div>\n
\n
\n
\u4f8b 2<\/h4>\n
$\\displaystyle \\int \\frac{dx}{\\sqrt{x^2+1}}$<\/p>\n
\u3053\u308c\u3082\u3059\u3050 integrate()<\/code> \u3067\u304d\u3066\u3057\u307e\u3044\u307e\u3059\u3002<\/p>\n<\/div>\n<\/div>\n<\/div>\n
\n
\n
In\u00a0[53]:<\/div>\n
\n
\n
\n
integrate<\/span>(<\/span>1<\/span>\/<\/span>sqrt<\/span>(<\/span>x<\/span>**<\/span>2<\/span> +<\/span> 1<\/span>),<\/span> x<\/span>)<\/span>\r\n<\/pre>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n
\n
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Out[53]:<\/div>\n
$\\displaystyle \\operatorname{asinh}{\\left(x \\right)}$<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n
\n
<\/div>\n
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\n
$\\log$ \u3067\u3042\u3089\u308f\u3059\u3068…<\/p>\n<\/div>\n<\/div>\n<\/div>\n
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In\u00a0[54]:<\/div>\n
\n
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_<\/span>.<\/span>rewrite<\/span>(<\/span>log<\/span>)<\/span>\r\n<\/pre>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n
\n
\n
\n
Out[54]:<\/div>\n
$\\displaystyle \\log{\\left(x + \\sqrt{x^{2} + 1} \\right)}$<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n
\n
<\/div>\n
\n
\n
\u30d2\u30f3\u30c8\u306b\u5f93\u3063\u3066\uff0c\u3042\u3048\u3066\u3053\u308c\u3092\u7f6e\u63db\u7a4d\u5206\u3057\u3066\u307f\u307e\u3059\u3002$x + \\sqrt{x^2 + 1} \\equiv t\\ (>0)$ \u3068\u304a\u304f\u3068…<\/p>\n<\/div>\n<\/div>\n<\/div>\n
\n
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In\u00a0[55]:<\/div>\n
\n
\n
\n
int1<\/span> =<\/span> Integral<\/span>(<\/span>1<\/span>\/<\/span>sqrt<\/span>(<\/span>x<\/span>**<\/span>2<\/span> +<\/span> 1<\/span>),<\/span> x<\/span>)<\/span>\r\nint1<\/span>\r\n<\/pre>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n
\n
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Out[55]:<\/div>\n
$\\displaystyle \\int \\frac{1}{\\sqrt{x^{2} + 1}}\\, dx$<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n
\n
\n
In\u00a0[56]:<\/div>\n
\n
\n
\n
# t \u306f\u6b63\u3067\u3042\u308b\u3068\u5ba3\u8a00<\/span>\r\nt<\/span> =<\/span> Symbol<\/span>(<\/span>'t'<\/span>,<\/span> positive<\/span> =<\/span> True<\/span>)<\/span>\r\n\r\nint2<\/span> =<\/span> int1<\/span>.<\/span>transform<\/span>(<\/span>x<\/span>+<\/span>sqrt<\/span>(<\/span>x<\/span>**<\/span>2<\/span>+<\/span>1<\/span>),<\/span> t<\/span>)<\/span>\r\nint2<\/span>\r\n<\/pre>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n
\n
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Out[56]:<\/div>\n
$\\displaystyle \\int \\frac{1 – \\frac{t^{2} – 1}{2 t^{2}}}{\\sqrt{1 + \\frac{\\left(t^{2} – 1\\right)^{2}}{4 t^{2}}}}\\, dt$<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n
\n
<\/div>\n
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\n
\u88ab\u7a4d\u5206\u95a2\u6570\u306f\u3082\u3063\u3068\u7c21\u5358\u306b\u306a\u308a\u307e\u3059\u3088\u306d\u3047\u3002<\/p>\n<\/div>\n<\/div>\n<\/div>\n
\n
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In\u00a0[57]:<\/div>\n
\n
\n
\n
# \u88ab\u7a4d\u5206\u95a2\u6570\u3092 ft \u3068\u3057\u3066\u53d6\u308a\u51fa\u3059<\/span>\r\nft<\/span> =<\/span> diff<\/span>(<\/span>int2<\/span>,<\/span> t<\/span>)<\/span>\r\nft<\/span>.<\/span>simplify<\/span>()<\/span>\r\n<\/pre>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n
\n
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Out[57]:<\/div>\n
$\\displaystyle \\frac{t^{2} + 1}{t \\sqrt{t^{4} + 2 t^{2} + 1}}$<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n
\n
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In\u00a0[58]:<\/div>\n
\n
\n
\n
# 2\u4e57\u3057\u3066\u7c21\u5358\u5316\u3057\u3066\u5e73\u65b9\u6839\u3092\u3068\u308b<\/span>\r\nft<\/span> =<\/span> sqrt<\/span>(<\/span>simplify<\/span>(<\/span>ft<\/span>**<\/span>2<\/span>))<\/span>\r\nft<\/span>\r\n<\/pre>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n
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Out[58]:<\/div>\n
$\\displaystyle \\frac{1}{t}$<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n
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In\u00a0[59]:<\/div>\n
\n
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\n
ans<\/span> =<\/span> integrate<\/span>(<\/span>ft<\/span>,<\/span> t<\/span>)<\/span>\r\nans<\/span>\r\n<\/pre>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n
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\n
\n
Out[59]:<\/div>\n
$\\displaystyle \\log{\\left(t \\right)}$<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n
\n
<\/div>\n
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\n
$t$ \u3092\u3082\u3068\u306e\u5909\u6570 $x + \\sqrt{x^2 + 1}$ \u306b\u306a\u304a\u3059\u3068…<\/p>\n<\/div>\n<\/div>\n<\/div>\n
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In\u00a0[60]:<\/div>\n
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ans<\/span>.<\/span>subs<\/span>(<\/span>t<\/span>,<\/span> x<\/span>+<\/span>sqrt<\/span>(<\/span>x<\/span>**<\/span>2<\/span>+<\/span>1<\/span>))<\/span>\r\n<\/pre>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n
\n
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Out[60]:<\/div>\n
$\\displaystyle \\log{\\left(x + \\sqrt{x^{2} + 1} \\right)}$<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n
\n
<\/div>\n
\n
\n
\u4f8b 3<\/h4>\n
$\\displaystyle \\int \\sqrt{x^2+1} dx$<\/p>\n
\u3053\u308c\u3082 integrate()<\/code> \u3067\u304d\u3066\u3057\u307e\u3044\u307e\u3059\u3002<\/p>\n<\/div>\n<\/div>\n<\/div>\n
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In\u00a0[61]:<\/div>\n
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integrate<\/span>(<\/span>sqrt<\/span>(<\/span>x<\/span>**<\/span>2<\/span> +<\/span> 1<\/span>),<\/span> x<\/span>)<\/span>\r\n<\/pre>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n
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Out[61]:<\/div>\n
$\\displaystyle \\frac{x \\sqrt{x^{2} + 1}}{2} + \\frac{\\operatorname{asinh}{\\left(x \\right)}}{2}$<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n
\n
<\/div>\n
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\n
$\\log$ \u3067\u3042\u3089\u308f\u3059\u3068…<\/p>\n<\/div>\n<\/div>\n<\/div>\n
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In\u00a0[62]:<\/div>\n
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_<\/span>.<\/span>rewrite<\/span>(<\/span>log<\/span>)<\/span>\r\n<\/pre>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n
\n
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Out[62]:<\/div>\n
$\\displaystyle \\frac{x \\sqrt{x^{2} + 1}}{2} + \\frac{\\log{\\left(x + \\sqrt{x^{2} + 1} \\right)}}{2}$<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n
\n
<\/div>\n
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\n
\u30d2\u30f3\u30c8\u306b\u3057\u305f\u304c\u3063\u3066\uff0c\u3053\u308c\u3092\u3042\u3048\u3066\u7f6e\u63db\u7a4d\u5206\u3067\u3084\u3063\u3066\u307f\u308b\u3002$x+\\sqrt{x^2+1} \\equiv t$ \u3068\u304a\u3044\u3066…<\/p>\n<\/div>\n<\/div>\n<\/div>\n
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In\u00a0[63]:<\/div>\n
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int1<\/span> =<\/span> Integral<\/span>(<\/span>sqrt<\/span>(<\/span>x<\/span>**<\/span>2<\/span> +<\/span> 1<\/span>),<\/span> x<\/span>)<\/span>\r\nint1<\/span>\r\n<\/pre>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n
\n
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Out[63]:<\/div>\n
$\\displaystyle \\int \\sqrt{x^{2} + 1}\\, dx$<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n
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In\u00a0[64]:<\/div>\n
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# t \u306f\u6b63\u3067\u3042\u308b\u3068\u5ba3\u8a00<\/span>\r\nt<\/span> =<\/span> Symbol<\/span>(<\/span>'t'<\/span>,<\/span> positive<\/span> =<\/span> True<\/span>)<\/span>\r\n\r\nint2<\/span> =<\/span> int1<\/span>.<\/span>transform<\/span>(<\/span>x<\/span>+<\/span>sqrt<\/span>(<\/span>x<\/span>**<\/span>2<\/span>+<\/span>1<\/span>),<\/span> t<\/span>)<\/span>\r\nint2<\/span>.<\/span>simplify<\/span>()<\/span>\r\n<\/pre>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n
\n
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Out[64]:<\/div>\n
$\\displaystyle \\int \\frac{\\left(t^{2} + 1\\right) \\sqrt{t^{4} + 2 t^{2} + 1}}{4 t^{3}}\\, dt$<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n
\n
<\/div>\n
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\u88ab\u7a4d\u5206\u95a2\u6570\u306f\u3082\u3063\u3068\u7c21\u5358\u306b\u306a\u308a\u307e\u3059\u3088\u306d\u3047\u3002<\/p>\n<\/div>\n<\/div>\n<\/div>\n
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In\u00a0[65]:<\/div>\n
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# \u88ab\u7a4d\u5206\u95a2\u6570\u3092 ft \u3068\u3057\u3066\u53d6\u308a\u51fa\u3059<\/span>\r\nft<\/span> =<\/span> diff<\/span>(<\/span>int2<\/span>.<\/span>simplify<\/span>(),<\/span> t<\/span>)<\/span>\r\nft<\/span>\r\n<\/pre>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n
\n
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Out[65]:<\/div>\n
$\\displaystyle \\frac{\\left(t^{2} + 1\\right) \\sqrt{t^{4} + 2 t^{2} + 1}}{4 t^{3}}$<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n
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In\u00a0[66]:<\/div>\n
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\n
\n
factor<\/span>(<\/span>ft<\/span>**<\/span>2<\/span>)<\/span>\r\n<\/pre>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n
\n
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Out[66]:<\/div>\n
$\\displaystyle \\frac{\\left(t^{2} + 1\\right)^{4}}{16 t^{6}}$<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n
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In\u00a0[67]:<\/div>\n
\n
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# 2\u4e57\u3057\u3066\u7c21\u5358\u5316\u3057\u3066\u5e73\u65b9\u6839\u3092\u3068\u308b<\/span>\r\nft<\/span> =<\/span> sqrt<\/span>(<\/span>factor<\/span>(<\/span>ft<\/span>**<\/span>2<\/span>))<\/span>\r\nft<\/span>.<\/span>apart<\/span>()<\/span>\r\n<\/pre>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n
\n
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Out[67]:<\/div>\n
$\\displaystyle \\frac{t}{4} + \\frac{1}{2 t} + \\frac{1}{4 t^{3}}$<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n
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In\u00a0[68]:<\/div>\n
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\n
\n
ans<\/span> =<\/span> integrate<\/span>(<\/span>ft<\/span>.<\/span>apart<\/span>(),<\/span> t<\/span>)<\/span>\r\nEq<\/span>(<\/span>Integral<\/span>(<\/span>ft<\/span>.<\/span>apart<\/span>(),<\/span> t<\/span>),<\/span> ans<\/span>)<\/span>\r\n<\/pre>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n
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Out[68]:<\/div>\n
$\\displaystyle \\int \\left(\\frac{t}{4} + \\frac{1}{2 t} + \\frac{1}{4 t^{3}}\\right)\\, dt = \\frac{t^{2}}{8} + \\frac{\\log{\\left(t \\right)}}{2} – \\frac{1}{8 t^{2}}$<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n
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<\/div>\n
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$t$ \u3092\u3082\u3068\u306e\u5909\u6570 $x+\\sqrt{x^2+1}$ \u306b\u623b\u3057\u3066…<\/p>\n<\/div>\n<\/div>\n<\/div>\n
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In\u00a0[69]:<\/div>\n
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ans<\/span>.<\/span>subs<\/span>(<\/span>t<\/span>,<\/span> x<\/span>+<\/span>sqrt<\/span>(<\/span>x<\/span>**<\/span>2<\/span>+<\/span>1<\/span>))<\/span>\r\n<\/pre>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n
\n
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Out[69]:<\/div>\n
$\\displaystyle \\frac{\\left(x + \\sqrt{x^{2} + 1}\\right)^{2}}{8} + \\frac{\\log{\\left(x + \\sqrt{x^{2} + 1} \\right)}}{2} – \\frac{1}{8 \\left(x + \\sqrt{x^{2} + 1}\\right)^{2}}$<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n
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<\/div>\n
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… \u3046\u301c\u3093\u3002\u306a\u304b\u306a\u304b\u307e\u3068\u3081\u3066\u304f\u308c\u307e\u305b\u3093\u306d\u3047\u3002<\/p>\n<\/div>\n<\/div>\n<\/div>\n
\n
<\/div>\n
\n
\n
\u7df4\u7fd2\u554f\u984c 1<\/h4>\n
$\\displaystyle \\int \\frac{dx}{x \\sqrt{x-1}}$<\/p>\n
\u76f4\u63a5 integrate()<\/code> \u3059\u308b\u3068…<\/p>\n<\/div>\n<\/div>\n<\/div>\n
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In\u00a0[70]:<\/div>\n
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integrate<\/span>(<\/span>1<\/span>\/<\/span>(<\/span>x<\/span>*<\/span>sqrt<\/span>(<\/span>x<\/span>-<\/span>1<\/span>)),<\/span> x<\/span>)<\/span>\r\n<\/pre>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n
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Out[70]:<\/div>\n
$\\displaystyle \\begin{cases} 2 i \\operatorname{acosh}{\\left(\\frac{1}{\\sqrt{x}} \\right)} & \\text{for}\\: \\frac{1}{\\left|{x}\\right|} > 1 \\\\- 2 \\operatorname{asin}{\\left(\\frac{1}{\\sqrt{x}} \\right)} & \\text{otherwise} \\end{cases}$<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n
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<\/div>\n
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\n
\u984c\u610f\u304b\u3089 $x>1$ \u3060\u304b\u3089\uff0c2\u884c\u76ee\u304c\u7b54\u3048\u306a\u3093\u3060\u308d\u3046\u3051\u3069\uff0c\u7a4d\u5206\u5b9a\u6570\u5206\u306e\u4e0d\u5b9a\u6027\u3092\u306e\u305e\u3044\u3066
\n$\\displaystyle 2 \\operatorname{acos} \\left(\\frac{1}{\\sqrt{x}}\\right)$ \u3068\u66f8\u3044\u305f\u307b\u3046\u304c\u3088\u308d\u3057\u3044\u304b\u3068\u601d\u3046\u3002<\/p>\n
SymPy, \u3061\u3087\u3063\u3068\u6b8b\u5ff5\u3002<\/p>\n
\u7f6e\u63db\u7a4d\u5206\u3067\u3084\u308b\u3068\uff0c$\\sqrt{x-1} \\equiv t$ \u3068\u304a\u3044\u3066…<\/p>\n<\/div>\n<\/div>\n<\/div>\n
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In\u00a0[71]:<\/div>\n
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int1<\/span> =<\/span> Integral<\/span>(<\/span>1<\/span>\/<\/span>(<\/span>x<\/span>*<\/span>sqrt<\/span>(<\/span>x<\/span>-<\/span>1<\/span>)),<\/span> x<\/span>)<\/span>\r\nint1<\/span>\r\n<\/pre>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n
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Out[71]:<\/div>\n
$\\displaystyle \\int \\frac{1}{x \\sqrt{x – 1}}\\, dx$<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n
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In\u00a0[72]:<\/div>\n
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int2<\/span> =<\/span> int1<\/span>.<\/span>transform<\/span>(<\/span>sqrt<\/span>(<\/span>x<\/span>-<\/span>1<\/span>),<\/span> t<\/span>)<\/span>\r\nint2<\/span>\r\n<\/pre>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n
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Out[72]:<\/div>\n
$\\displaystyle \\int \\frac{2}{t^{2} + 1}\\, dt$<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n
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In\u00a0[73]:<\/div>\n
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ans<\/span> =<\/span> int2<\/span>.<\/span>doit<\/span>()<\/span>\r\nans<\/span>\r\n<\/pre>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n
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Out[73]:<\/div>\n
$\\displaystyle 2 \\operatorname{atan}{\\left(t \\right)}$<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n
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In\u00a0[74]:<\/div>\n
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ans<\/span>.<\/span>subs<\/span>(<\/span>t<\/span>,<\/span> sqrt<\/span>(<\/span>x<\/span>-<\/span>1<\/span>))<\/span>\r\n<\/pre>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n
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Out[74]:<\/div>\n
$\\displaystyle 2 \\operatorname{atan}{\\left(\\sqrt{x – 1} \\right)}$<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n
\n
<\/div>\n
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\n
\u3053\u306e\u7b54\u3048\u304c\u76f4\u63a5 integrate()<\/code> \u3067\u6c42\u3081\u305f $\\displaystyle 2 \\operatorname{acos}{\\left(\\frac{1}{\\sqrt{x}}\\right)}$ \u3068\u540c\u7b49\uff0c\u3059\u306a\u308f\u3061<\/p>\n
$$\\tan^{-1} \\sqrt{x-1} = \\cos^{-1} \\frac{1}{\\sqrt{x}}$$<\/p>\n
\u3067\u3042\u308b\u3053\u3068\u3092\u78ba\u8a8d\u3057\u3066\u307f\u308b\u3002$\\tan^{-1}$ \u306e\u5f0f\u3092 $\\cos^{-1}$ \u3092\u4f7f\u3063\u3066\u66f8\u304d\u76f4\u3059\u3068…<\/p>\n<\/div>\n<\/div>\n<\/div>\n
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In\u00a0[75]:<\/div>\n
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Eq<\/span>(<\/span>atan<\/span>(<\/span>sqrt<\/span>(<\/span>x<\/span>-<\/span>1<\/span>)),<\/span> \r\n   atan<\/span>(<\/span>sqrt<\/span>(<\/span>x<\/span>-<\/span>1<\/span>))<\/span>.<\/span>rewrite<\/span>(<\/span>acos<\/span>))<\/span>\r\n<\/pre>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n
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Out[75]:<\/div>\n
$\\displaystyle \\operatorname{atan}{\\left(\\sqrt{x – 1} \\right)} = \\operatorname{acos}{\\left(\\frac{1}{\\sqrt{x}} \\right)}$<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n
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In\u00a0[76]:<\/div>\n
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Eq<\/span>(<\/span>acos<\/span>(<\/span>1<\/span>\/<\/span>sqrt<\/span>(<\/span>x<\/span>)),<\/span> \r\n   acos<\/span>(<\/span>1<\/span>\/<\/span>sqrt<\/span>(<\/span>x<\/span>))<\/span>.<\/span>rewrite<\/span>(<\/span>atan<\/span>))<\/span>\r\n<\/pre>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n
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Out[76]:<\/div>\n
$\\displaystyle \\operatorname{acos}{\\left(\\frac{1}{\\sqrt{x}} \\right)} = \\operatorname{atan}{\\left(\\sqrt{x} \\sqrt{1 – \\frac{1}{x}} \\right)}$<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n
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SymPy \u306f\u5e73\u65b9\u6839\u540c\u58eb\u306e\u304b\u3051\u7b97\u304c\u82e6\u624b\u306e\u3088\u3046\u3067\u3042\u308b\u3002<\/p>\n
$$\\sqrt{x} \\sqrt{1 – \\frac{1}{{x}}} = \\sqrt{x-1}$$<\/p>\n
\u3060\u3088\u306d\u3047\u3002<\/p>\n<\/div>\n<\/div>\n<\/div>\n
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\u7df4\u7fd2\u554f\u984c 2<\/h4>\n
$\\displaystyle \\int \\frac{\\sqrt{x}}{\\sqrt{1-x}} dx$<\/p>\n<\/div>\n<\/div>\n<\/div>\n
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In\u00a0[77]:<\/div>\n
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integrate<\/span>(<\/span>sqrt<\/span>(<\/span>x<\/span>)<\/span>\/<\/span>sqrt<\/span>(<\/span>1<\/span>-<\/span>x<\/span>),<\/span> x<\/span>)<\/span>\r\n<\/pre>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n
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Out[77]:<\/div>\n
$\\displaystyle \\begin{cases} – i \\sqrt{x} \\sqrt{x – 1} – i \\operatorname{acosh}{\\left(\\sqrt{x} \\right)} & \\text{for}\\: \\left|{x}\\right| > 1 \\\\\\frac{x^{\\frac{3}{2}}}{\\sqrt{1 – x}} – \\frac{\\sqrt{x}}{\\sqrt{1 – x}} + \\operatorname{asin}{\\left(\\sqrt{x} \\right)} & \\text{otherwise} \\end{cases}$<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n
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\u984c\u610f\u304b\u3089 $0 \\leq x < 1$ \u306a\u306e\u30672\u884c\u76ee\u304c\u7b54\u3048\u306a\u3093\u3060\u3051\u3069\uff0c\u3053\u308c\u3082\u7f6e\u63db\u7a4d\u5206\u3067\u5225\u9014\u8a08\u7b97\u3057\u3066\u307f\u308b\u3002<\/p>\n
$\\displaystyle \\frac{\\sqrt{x}}{\\sqrt{1-x}} \\equiv t$ \u3068\u304a\u3044\u3066…<\/p>\n<\/div>\n<\/div>\n<\/div>\n
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In\u00a0[78]:<\/div>\n
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int1<\/span> =<\/span> Integral<\/span>(<\/span>sqrt<\/span>(<\/span>x<\/span>)<\/span>\/<\/span>sqrt<\/span>(<\/span>1<\/span>-<\/span>x<\/span>),<\/span> x<\/span>)<\/span>\r\nint1<\/span>\r\n<\/pre>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n
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Out[78]:<\/div>\n
$\\displaystyle \\int \\frac{\\sqrt{x}}{\\sqrt{1 – x}}\\, dx$<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n
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In\u00a0[79]:<\/div>\n
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t<\/span> =<\/span> Symbol<\/span>(<\/span>'t'<\/span>,<\/span> positive<\/span> =<\/span> True<\/span>)<\/span>\r\n\r\nint2<\/span> =<\/span> int1<\/span>.<\/span>transform<\/span>(<\/span>sqrt<\/span>(<\/span>x<\/span>)<\/span>\/<\/span>sqrt<\/span>(<\/span>1<\/span>-<\/span>x<\/span>),<\/span> t<\/span>)<\/span>\r\nint2<\/span>\r\n<\/pre>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n
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Out[79]:<\/div>\n
$\\displaystyle \\int \\frac{t \\left(- \\frac{2 t^{3}}{\\left(t^{2} + 1\\right)^{2}} + \\frac{2 t}{t^{2} + 1}\\right)}{\\sqrt{t^{2} + 1} \\sqrt{- \\frac{t^{2}}{t^{2} + 1} + 1}}\\, dt$<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n
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In\u00a0[80]:<\/div>\n
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# \u88ab\u7a4d\u5206\u95a2\u6570\u3092 ft \u3068\u3057\u3066\u3068\u308b<\/span>\r\nft<\/span> =<\/span> diff<\/span>(<\/span>int2<\/span>,<\/span> t<\/span>)<\/span>\r\n# 2\u4e57\u3057\u3066\u7c21\u5358\u5316\u3057\u3066\u5e73\u65b9\u6839\u3092\u3068\u308b<\/span>\r\nft<\/span> =<\/span> sqrt<\/span>((<\/span>ft<\/span>**<\/span>2<\/span>)<\/span>.<\/span>simplify<\/span>())<\/span>\r\nft<\/span>\r\n<\/pre>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n
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Out[80]:<\/div>\n
$\\displaystyle \\frac{2 t^{2}}{\\left(t^{2} + 1\\right)^{2}}$<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n
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In\u00a0[81]:<\/div>\n
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ans<\/span> =<\/span> integrate<\/span>(<\/span>ft<\/span>,<\/span> t<\/span>)<\/span>\r\nans<\/span>\r\n<\/pre>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n
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Out[81]:<\/div>\n
$\\displaystyle – \\frac{2 t}{2 t^{2} + 2} + \\operatorname{atan}{\\left(t \\right)}$<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n
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In\u00a0[82]:<\/div>\n
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ans<\/span>.<\/span>subs<\/span>(<\/span>t<\/span>,<\/span> sqrt<\/span>(<\/span>x<\/span>)<\/span>\/<\/span>sqrt<\/span>(<\/span>1<\/span>-<\/span>x<\/span>))<\/span>.<\/span>simplify<\/span>()<\/span>\r\n<\/pre>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n
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Out[82]:<\/div>\n
$\\displaystyle \\frac{\\sqrt{x} \\left(x – 1\\right) + \\sqrt{1 – x} \\operatorname{atan}{\\left(\\frac{\\sqrt{x}}{\\sqrt{1 – x}} \\right)}}{\\sqrt{1 – x}}$<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n
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\u4eba\u9593\u306e\u76ee\u3067\u306f\uff0c<\/p>\n
$\\displaystyle \\operatorname{atan}{\\left(\\frac{\\sqrt{x}}{\\sqrt{1 – x}} \\right)} – \\sqrt{x}\\sqrt{1-x}$<\/p>\n
\u3068\u3057\u305f\u307b\u3046\u304c\u898b\u3084\u3059\u3044\u304b\u3082\u3002<\/p>\n<\/div>\n<\/div>\n<\/div>\n
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\u7df4\u7fd2\u554f\u984c 3<\/h4>\n
$\\displaystyle \\int \\frac{\\sqrt{x}}{\\sqrt{1-x^3}} dx$<\/p>\n<\/div>\n<\/div>\n<\/div>\n
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In\u00a0[83]:<\/div>\n
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integrate<\/span>(<\/span>sqrt<\/span>(<\/span>x<\/span>)<\/span>\/<\/span>sqrt<\/span>(<\/span>1<\/span>-<\/span>x<\/span>**<\/span>3<\/span>),<\/span> x<\/span>)<\/span>\r\n<\/pre>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n
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Out[83]:<\/div>\n
$\\displaystyle \\begin{cases} – \\frac{2 i \\operatorname{acosh}{\\left(x^{\\frac{3}{2}} \\right)}}{3} & \\text{for}\\: \\left|{x^{3}}\\right| > 1 \\\\\\frac{2 \\operatorname{asin}{\\left(x^{\\frac{3}{2}} \\right)}}{3} & \\text{otherwise} \\end{cases}$<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n
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\u984c\u610f\u304b\u3089 $|x^3| < 1$ \u306a\u306e\u30672\u884c\u76ee\u304c\u7b54\u3048\u3002<\/p>\n<\/div>\n<\/div>\n<\/div>\n
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\u3061\u306a\u307f\u306b\uff0c$y = \\sin^{-1} x^{3\/2}$ \u3068\u304a\u304f\u3068\uff0c<\/p>\n
\\begin{eqnarray}
\nx^{\\frac{3}{2}} &=& \\sin y \\\\
\n\\tan y &=& \\frac{\\sin y}{\\cos y} \\\\
\n&=& \\frac{\\sin y}{\\sqrt{1-\\sin^2 y}} \\\\
\n&=& \\frac{x^{\\frac{3}{2}}}{\\sqrt{1-x^{3}}} \\\\
\n\\therefore\\ \\ y &=& \\tan^{-1} \\frac{\\sqrt{x^3}}{\\sqrt{1-x^{3}}}
\n\\end{eqnarray}<\/p>\n<\/div>\n<\/div>\n<\/div>\n
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\u7df4\u7fd2\u554f\u984c 4<\/h4>\n
$\\displaystyle \\int \\frac{1}{\\left(a^2 +x^2\\right)^{\\frac{3}{2}}} dx$<\/p>\n<\/div>\n<\/div>\n<\/div>\n
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In\u00a0[84]:<\/div>\n
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integrate<\/span>(<\/span>1<\/span>\/<\/span>sqrt<\/span>(<\/span>a<\/span>**<\/span>2<\/span> +<\/span> x<\/span>**<\/span>2<\/span>)<\/span>**<\/span>3<\/span>,<\/span> x<\/span>)<\/span>\r\n<\/pre>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n
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Out[84]:<\/div>\n
$\\displaystyle \\frac{x}{a^{2} \\sqrt{a^{2} + x^{2}}}$<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n
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\u5e83\u7fa9\u306e\u7a4d\u5206<\/h3>\n
SymPy \u3067\u306f\u7121\u9650\u5927 $\\infty$ \u306f oo<\/code> (\u30a2\u30eb\u30d5\u30a1\u30d9\u30c3\u30c8\u306e\u30aa\u30fc\u306e\u5c0f\u6587\u5b572\u3064) \u3067\u3059\u3002<\/p>\n<\/div>\n<\/div>\n<\/div>\n
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\u7df4\u7fd2\u554f\u984c 1<\/h4>\n
$\\displaystyle \\int_{-\\infty}^{\\infty} \\frac{1}{(a^2 + x^2)^{\\frac{3}{2}}} dx$<\/p>\n<\/div>\n<\/div>\n<\/div>\n
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In\u00a0[85]:<\/div>\n
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a<\/span> =<\/span> Symbol<\/span>(<\/span>'a'<\/span>,<\/span> positive<\/span> =<\/span> True<\/span>)<\/span>\r\n\r\nEq<\/span>(<\/span>Integral<\/span>(<\/span>1<\/span>\/<\/span>sqrt<\/span>(<\/span>a<\/span>**<\/span>2<\/span> +<\/span> x<\/span>**<\/span>2<\/span>)<\/span>**<\/span>3<\/span>,<\/span> (<\/span>x<\/span>,<\/span> -<\/span>oo<\/span>,<\/span> oo<\/span>)),<\/span>\r\n   Integral<\/span>(<\/span>1<\/span>\/<\/span>sqrt<\/span>(<\/span>a<\/span>**<\/span>2<\/span> +<\/span> x<\/span>**<\/span>2<\/span>)<\/span>**<\/span>3<\/span>,<\/span> (<\/span>x<\/span>,<\/span> -<\/span>oo<\/span>,<\/span> oo<\/span>))<\/span>.<\/span>doit<\/span>())<\/span>\r\n<\/pre>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n
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Out[85]:<\/div>\n
$\\displaystyle \\int\\limits_{-\\infty}^{\\infty} \\frac{1}{\\left(a^{2} + x^{2}\\right)^{\\frac{3}{2}}}\\, dx = \\frac{2}{a^{2}}$<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n
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<\/div>\n
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\u7df4\u7fd2\u554f\u984c 2<\/h4>\n
$\\displaystyle \\int_{-\\infty}^{\\infty} \\frac{1}{a^2 + x^2} dx$<\/p>\n<\/div>\n<\/div>\n<\/div>\n
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In\u00a0[86]:<\/div>\n
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Eq<\/span>(<\/span>Integral<\/span>(<\/span>1<\/span>\/<\/span>(<\/span>a<\/span>**<\/span>2<\/span>+<\/span>x<\/span>**<\/span>2<\/span>),<\/span> (<\/span>x<\/span>,<\/span> -<\/span>oo<\/span>,<\/span> oo<\/span>)),<\/span>\r\n   Integral<\/span>(<\/span>1<\/span>\/<\/span>(<\/span>a<\/span>**<\/span>2<\/span>+<\/span>x<\/span>**<\/span>2<\/span>),<\/span> (<\/span>x<\/span>,<\/span> -<\/span>oo<\/span>,<\/span> oo<\/span>))<\/span>.<\/span>doit<\/span>())<\/span>\r\n<\/pre>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n
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Out[86]:<\/div>\n
$\\displaystyle \\int\\limits_{-\\infty}^{\\infty} \\frac{1}{a^{2} + x^{2}}\\, dx = \\frac{\\pi}{a}$<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n
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\u3044\u304f\u3064\u304b\u306e\u5fdc\u7528<\/h3>\n
\u3044\u304f\u3064\u304b\u306e\u5fdc\u7528\u306e\u9805\u306e\u4f8b\u984c\u3002<\/p>\n<\/div>\n<\/div>\n<\/div>\n
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\u5186\u306e\u9762\u7a4d<\/h4>\n
$\\displaystyle x^2 + y^2 = r^2$ \u3088\u308a $y = \\sqrt{r^2 – x^2}$\uff08\u5186\u306e\u4e0a\u534a\u5206\uff09\u3002 \u3053\u3053\u3067 $r$ \u306f\u5186\u306e\u534a\u5f84\u3067 $r > 0$<\/p>\n
\u5186\u306e\u9762\u7a4d $S$ \u306f\uff0c$y = \\sqrt{r^2 – x^2}$ \u3068 $x$ \u8ef8\u3067\u56f2\u307e\u308c\u308b\u90e8\u5206\u306e\u9762\u7a4d\u3092\u6c42\u3081\u30662\u500d\u3059\u308c\u3070\u3088\u3044\u3002<\/p>\n
$\\displaystyle S = 2 \\int_{-r}^r y\\, dx$<\/p>\n<\/div>\n<\/div>\n<\/div>\n
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In\u00a0[87]:<\/div>\n
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r<\/span> =<\/span> Symbol<\/span>(<\/span>'r'<\/span>,<\/span> positive<\/span> =<\/span> True<\/span>)<\/span>\r\n\r\ny<\/span> =<\/span> sqrt<\/span>(<\/span>r<\/span>**<\/span>2<\/span> -<\/span> x<\/span>**<\/span>2<\/span>)<\/span>\r\n\r\nEq<\/span>(<\/span>Integral<\/span>(<\/span>2<\/span>*<\/span>y<\/span>,<\/span> (<\/span>x<\/span>,<\/span> -<\/span>r<\/span>,<\/span> r<\/span>)),<\/span> \r\n   Integral<\/span>(<\/span>2<\/span>*<\/span>y<\/span>,<\/span> (<\/span>x<\/span>,<\/span> -<\/span>r<\/span>,<\/span> r<\/span>))<\/span>.<\/span>doit<\/span>())<\/span>\r\n<\/pre>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n
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Out[87]:<\/div>\n
$\\displaystyle \\int\\limits_{- r}^{r} 2 \\sqrt{r^{2} – x^{2}}\\, dx = \\pi r^{2}$<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n
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\u5186\u5468<\/h4>\n
$\\displaystyle L = 2 \\int_{-r}^r \\sqrt{1 + \\left(\\frac{dy}{dx}\\right)^2}\\, dx$<\/p>\n<\/div>\n<\/div>\n<\/div>\n
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In\u00a0[88]:<\/div>\n
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dydx<\/span> =<\/span> diff<\/span>(<\/span>y<\/span>,<\/span> x<\/span>)<\/span>\r\ndydx<\/span>\r\n<\/pre>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n
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Out[88]:<\/div>\n
$\\displaystyle – \\frac{x}{\\sqrt{r^{2} – x^{2}}}$<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n
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In\u00a0[89]:<\/div>\n
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Eq<\/span>(<\/span>2<\/span>*<\/span>Integral<\/span>(<\/span>sqrt<\/span>(<\/span>1<\/span>+<\/span>dydx<\/span>**<\/span>2<\/span>),<\/span> (<\/span>x<\/span>,<\/span> -<\/span>r<\/span>,<\/span> r<\/span>)),<\/span>\r\n   2<\/span>*<\/span>Integral<\/span>(<\/span>sqrt<\/span>(<\/span>1<\/span>+<\/span>dydx<\/span>**<\/span>2<\/span>),<\/span> (<\/span>x<\/span>,<\/span> -<\/span>r<\/span>,<\/span> r<\/span>))<\/span>.<\/span>doit<\/span>())<\/span>\r\n<\/pre>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n
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Out[89]:<\/div>\n
$\\displaystyle 2 \\int\\limits_{- r}^{r} \\sqrt{\\frac{x^{2}}{r^{2} – x^{2}} + 1}\\, dx = 2 \\pi r$<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n
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\u7403\u306e\u8868\u9762\u7a4d<\/h4>\n
$\\displaystyle S = \\int_{-r}^r 2\\pi y \\sqrt{1 + \\left(\\frac{dy}{dx}\\right)^2}\\, dx$<\/p>\n
SymPy \u306f\u5e73\u65b9\u6839\u304c\u3042\u308b\u3053\u306e\u624b\u306e\u7a4d\u5206\u304c\u82e6\u624b\u3089\u3057\u3044\u306e\u3067\uff0c\u88ab\u7a4d\u5206\u95a2\u6570\u3092\u7c21\u5358\u5316\u3057\u3066\u304b\u3089\u8a08\u7b97\u3059\u308b\u3002<\/p>\n<\/div>\n<\/div>\n<\/div>\n
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In\u00a0[90]:<\/div>\n
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int1<\/span> =<\/span> Integral<\/span>(<\/span>2<\/span>*<\/span>pi<\/span>*<\/span>y<\/span>*<\/span>sqrt<\/span>(<\/span>1<\/span>+<\/span>diff<\/span>(<\/span>y<\/span>,<\/span>x<\/span>)<\/span>**<\/span>2<\/span>),<\/span> x<\/span>)<\/span>\r\nint1<\/span>\r\n<\/pre>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n
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Out[90]:<\/div>\n
$\\displaystyle \\int 2 \\pi \\sqrt{r^{2} – x^{2}} \\sqrt{\\frac{x^{2}}{r^{2} – x^{2}} + 1}\\, dx$<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n
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In\u00a0[91]:<\/div>\n
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fx<\/span> =<\/span> diff<\/span>(<\/span>int1<\/span>,<\/span> x<\/span>)<\/span>\r\nfx<\/span>\r\n<\/pre>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n
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Out[91]:<\/div>\n
$\\displaystyle 2 \\pi \\sqrt{r^{2} – x^{2}} \\sqrt{\\frac{x^{2}}{r^{2} – x^{2}} + 1}$<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n
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In\u00a0[92]:<\/div>\n
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# \u88ab\u7a4d\u5206\u95a2\u6570\u30922\u4e57\u3057\u3066\u7c21\u5358\u5316\u3057\u3066\u5e73\u65b9\u6839\u3092\u3068\u308b<\/span>\r\nfx<\/span> =<\/span> sqrt<\/span>((<\/span>fx<\/span>**<\/span>2<\/span>)<\/span>.<\/span>simplify<\/span>())<\/span>\r\nfx<\/span>\r\n<\/pre>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n
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Out[92]:<\/div>\n
$\\displaystyle 2 \\pi r$<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n
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In\u00a0[93]:<\/div>\n
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integrate<\/span>(<\/span>fx<\/span>,<\/span> (<\/span>x<\/span>,<\/span> -<\/span>r<\/span>,<\/span> r<\/span>))<\/span>\r\n<\/pre>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n
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Out[93]:<\/div>\n
$\\displaystyle 4 \\pi r^{2}$<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n
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\u7403\u306e\u4f53\u7a4d<\/h4>\n
$\\displaystyle V = \\int_{-r}^r \\pi y^2\\, dx$<\/p>\n<\/div>\n<\/div>\n<\/div>\n
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In\u00a0[94]:<\/div>\n
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integrate<\/span>(<\/span>pi<\/span>*<\/span>y<\/span>**<\/span>2<\/span>,<\/span> (<\/span>x<\/span>,<\/span> -<\/span>r<\/span>,<\/span> r<\/span>))<\/span>\r\n<\/pre>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n
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Out[94]:<\/div>\n
$\\displaystyle \\frac{4 \\pi r^{3}}{3}$<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n","protected":false},"excerpt":{"rendered":"","protected":false},"author":33,"featured_media":0,"parent":6034,"menu_order":30,"comment_status":"closed","ping_status":"closed","template":"","meta":{"inline_featured_image":false,"footnotes":""},"class_list":["post-6069","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\/6069","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=6069"}],"version-history":[{"count":6,"href":"https:\/\/home.hirosaki-u.ac.jp\/relativity\/wp-json\/wp\/v2\/pages\/6069\/revisions"}],"predecessor-version":[{"id":8090,"href":"https:\/\/home.hirosaki-u.ac.jp\/relativity\/wp-json\/wp\/v2\/pages\/6069\/revisions\/8090"}],"up":[{"embeddable":true,"href":"https:\/\/home.hirosaki-u.ac.jp\/relativity\/wp-json\/wp\/v2\/pages\/6034"}],"wp:attachment":[{"href":"https:\/\/home.hirosaki-u.ac.jp\/relativity\/wp-json\/wp\/v2\/media?parent=6069"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}