{"id":2197,"date":"2023-04-10T11:47:24","date_gmt":"2023-04-10T02:47:24","guid":{"rendered":"https:\/\/home.hirosaki-u.ac.jp\/relativity\/?page_id=2197"},"modified":"2023-04-10T11:47:57","modified_gmt":"2023-04-10T02:47:57","slug":"maxima-%e3%81%a6%e3%82%99-1-%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\/maxima-%e3%81%a7%e7%90%86%e5%b7%a5%e7%b3%bb%e3%81%ae%e6%95%b0%e5%ad%a6b\/maxima-%e3%81%a6%e3%82%99-1-%e5%a4%89%e6%95%b0%e9%96%a2%e6%95%b0%e3%81%ae%e7%a9%8d%e5%88%86\/","title":{"rendered":"Maxima \u3066\u3099 1 \u5909\u6570\u95a2\u6570\u306e\u7a4d\u5206"},"content":{"rendered":"

<\/p>\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\u3066\u3044\u308b\u3002<\/p>\n

Maxima \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$p \\neq -1$ \u306e\u4eee\u5b9a\u3092\u3057\u3066\u304b\u3089\u7a4d\u5206\u3057\u307e\u3059\u3002<\/p>\n<\/div>\n<\/div>\n<\/div>\n

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In\u00a0[1]:<\/div>\n
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assume<\/span>(<\/span>notequal<\/span>(<\/span>p<\/span>, -<\/span>1<\/span>))<\/span>;\r\n<\/pre>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n
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Out[1]:<\/div>\n
\\[\\tag{${\\it \\%o}_{1}$}\\left[ {\\it notequal}\\left(p , -1\\right) \\right] \\]<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n
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In\u00a0[2]:<\/div>\n
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integrate<\/span>(<\/span>x<\/span>**<\/span>p<\/span>, x<\/span>)<\/span>;\r\n<\/pre>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n
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Out[2]:<\/div>\n
\\[\\tag{${\\it \\%o}_{2}$}\\frac{x^{p+1}}{p+1}\\]<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n
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\u53c2\u8003\u307e\u3067\u306b\uff0c$p \\neq -1$ \u306e\u4eee\u5b9a assume(notequal(p, -1))$<\/code> \u3092\u3057\u306a\u3044\u3067 $\\displaystyle \\int x^p dx$ \u3092\u3055\u305b\u308b\u3068\u3069\u3046\u306a\u308b\u304b\uff0c\u8a66\u3057\u307e\u3059\u3002<\/p>\n<\/div>\n<\/div>\n<\/div>\n
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In\u00a0[3]:<\/div>\n
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\/* facts(p) \u3067 p \u306b\u5bfe\u3059\u308b\u4eee\u5b9a\u3092\u8868\u793a\u3055\u305b... *\/<\/span>\r\n\/* forget(%) \u3067\u8868\u793a\u3055\u305b\u305f\u4eee\u5b9a\u3092\u5fd8\u308c\u3055\u305b\u307e\u3059\u3002 *\/<\/span>\r\nfacts<\/span>(<\/span>p<\/span>)<\/span>$\r\nforget<\/span>(<\/span>%<\/span>)<\/span>$\r\n\/* \u3053\u308c\u3067\uff0cp \u306b\u5bfe\u3059\u308b\u4eee\u5b9a\u306f\u306a\u304b\u3063\u305f\u3053\u3068\u306b\u306a\u308a\u307e\u3059\u3002*\/<\/span>\r\n<\/pre>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n
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In\u00a0[4]:<\/div>\n
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integrate<\/span>(<\/span>x<\/span>**<\/span>p<\/span>, x<\/span>)<\/span>;\r\n\r\n\/* no; \u3068\u7b54\u3048\u307e\u3059\u3002*\/<\/span>\r\n<\/pre>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n
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\u001bXIs p equal to - 1?\r\n\u001b\\Is p equal to - 1?\r\nno;\r\n\r\n<\/pre>\n<\/div>\n<\/div>\n
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Out[4]:<\/div>\n
\\[\\tag{${\\it \\%o}_{5}$}\\frac{x^{p+1}}{p+1}\\]<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n
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In\u00a0[5]:<\/div>\n
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integrate<\/span>(<\/span>x<\/span>**<\/span>p<\/span>, x<\/span>)<\/span>;\r\n\r\n\/* \u4eca\u5ea6\u306f yes; \u3068\u7b54\u3048\u307e\u3059\u3002*\/<\/span>\r\n<\/pre>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n
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<\/div>\n
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\u001bXIs p equal to - 1?\r\n\u001b\\Is p equal to - 1?\r\nyes;\r\n\r\n<\/pre>\n<\/div>\n<\/div>\n
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Out[5]:<\/div>\n
\\[\\tag{${\\it \\%o}_{6}$}\\log x\\]<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n
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\u6307\u6570\u95a2\u6570<\/h4>\n
$\\displaystyle \\int e^x\\, dx$<\/p>\n<\/div>\n<\/div>\n<\/div>\n
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In\u00a0[6]:<\/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>exp<\/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[6]:<\/div>\n
\\[\\tag{${\\it \\%o}_{7}$}e^{x}\\]<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n
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In\u00a0[7]:<\/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\n'<\/span>integrate<\/span>(<\/span>exp<\/span>(<\/span>x<\/span>)<\/span>, x<\/span>)<\/span> =<\/span>\r\n integrate<\/span>(<\/span>exp<\/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[7]:<\/div>\n
\\[\\tag{${\\it \\%o}_{8}$}\\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[8]:<\/div>\n
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'<\/span>integrate<\/span>(<\/span>1\/<\/span>x<\/span>, x<\/span>)<\/span> =<\/span> \r\n integrate<\/span>(<\/span>1\/<\/span>x<\/span>, x<\/span>)<\/span>;\r\n<\/pre>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n
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Out[8]:<\/div>\n
\\[\\tag{${\\it \\%o}_{9}$}\\int {\\frac{1}{x}}{\\;dx}=\\log x\\]<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n
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Maxima \u306e\u7a4d\u5206\u3067\u306f $\\log$ \u306e\u4e2d\u8eab\u306e\u7d76\u5bfe\u5024\u3092\u7701\u7565\u3059\u308b\u3088\u3046\u3060\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[9]:<\/div>\n
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'<\/span>integrate<\/span>(<\/span>cos<\/span>(<\/span>x<\/span>)<\/span>, x<\/span>)<\/span> =<\/span> integrate<\/span>(<\/span>cos<\/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[9]:<\/div>\n
\\[\\tag{${\\it \\%o}_{10}$}\\int {\\cos x}{\\;dx}=\\sin x\\]<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n
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In\u00a0[10]:<\/div>\n
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'<\/span>integrate<\/span>(<\/span>sin<\/span>(<\/span>x<\/span>)<\/span>, x<\/span>)<\/span> =<\/span> integrate<\/span>(<\/span>sin<\/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[10]:<\/div>\n
\\[\\tag{${\\it \\%o}_{11}$}\\int {\\sin x}{\\;dx}=-\\cos x\\]<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n
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In\u00a0[11]:<\/div>\n
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'<\/span>integrate<\/span>(<\/span>1\/<\/span>(<\/span>cos<\/span>(<\/span>x<\/span>))<\/span>**<\/span>2<\/span>, x<\/span>)<\/span> =<\/span> integrate<\/span>(<\/span>1\/<\/span>(<\/span>cos<\/span>(<\/span>x<\/span>))<\/span>**<\/span>2<\/span>, x<\/span>)<\/span>;\r\n<\/pre>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n
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Out[11]:<\/div>\n
\\[\\tag{${\\it \\%o}_{12}$}\\int {\\frac{1}{\\cos ^2x}}{\\;dx}=\\tan x\\]<\/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[12]:<\/div>\n
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'<\/span>integrate<\/span>(<\/span>1\/<\/span>sqrt<\/span>(<\/span>1-<\/span>x<\/span>**<\/span>2<\/span>)<\/span>, x<\/span>)<\/span> =<\/span> \r\n integrate<\/span>(<\/span>1\/<\/span>sqrt<\/span>(<\/span>1-<\/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[12]:<\/div>\n
\\[\\tag{${\\it \\%o}_{13}$}\\int {\\frac{1}{\\sqrt{1-x^2}}}{\\;dx}=\\arcsin x\\]<\/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[13]:<\/div>\n
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'<\/span>integrate<\/span>(<\/span>-<\/span>1\/<\/span>sqrt<\/span>(<\/span>1-<\/span>x<\/span>**<\/span>2<\/span>)<\/span>, x<\/span>)<\/span> =<\/span> \r\n integrate<\/span>(<\/span>-<\/span>1\/<\/span>sqrt<\/span>(<\/span>1-<\/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[13]:<\/div>\n
\\[\\tag{${\\it \\%o}_{14}$}-\\int {\\frac{1}{\\sqrt{1-x^2}}}{\\;dx}=-\\arcsin x\\]<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n
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\u5b9f\u306f
\n$$\\cos^{-1} x + \\sin^{-1} x = \\frac{\\pi}{2}$$\u3068\u3044\u3046\u95a2\u4fc2\u304c\u3042\u3063\u305f\u306e\u3067\uff0c
\n$$\\cos^{-1} x = – \\sin^{-1} x + \\frac{\\pi}{2}$$<\/p>\n
$\\cos^{-1} x$ \u3068 $- \\sin^{-1} x$ \u306e\u9055\u3044\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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In\u00a0[14]:<\/div>\n
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'<\/span>integrate<\/span>(<\/span>1\/<\/span>(<\/span>1+<\/span>x<\/span>**<\/span>2<\/span>)<\/span>, x<\/span>)<\/span> =<\/span> \r\n integrate<\/span>(<\/span>1\/<\/span>(<\/span>1+<\/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[14]:<\/div>\n
\\[\\tag{${\\it \\%o}_{15}$}\\int {\\frac{1}{x^2+1}}{\\;dx}=\\arctan x\\]<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n
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\u5b9a\u7a4d\u5206<\/h3>\n
Maxima \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
\u4f8b\uff1a<\/p>\n<\/div>\n<\/div>\n<\/div>\n
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In\u00a0[15]:<\/div>\n
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'<\/span>integrate<\/span>(<\/span>x<\/span>**<\/span>2<\/span>, x<\/span>, a<\/span>, b<\/span>)<\/span> =<\/span>\r\n integrate<\/span>(<\/span>x<\/span>**<\/span>2<\/span>, x<\/span>, a<\/span>, b<\/span>)<\/span>;\r\n<\/pre>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n
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Out[15]:<\/div>\n
\\[\\tag{${\\it \\%o}_{16}$}\\int_{a}^{b}{x^2\\;dx}=\\frac{b^3}{3}-\\frac{a^3}{3}\\]<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n
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\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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In\u00a0[16]:<\/div>\n
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'<\/span>integrate<\/span>(<\/span>log<\/span>(<\/span>x<\/span>)<\/span>\/<\/span>x<\/span>, x<\/span>)<\/span> =<\/span>\r\n integrate<\/span>(<\/span>log<\/span>(<\/span>x<\/span>)<\/span>\/<\/span>x<\/span>, x<\/span>)<\/span>;\r\n<\/pre>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n
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Out[16]:<\/div>\n
\\[\\tag{${\\it \\%o}_{17}$}\\int {\\frac{\\log x}{x}}{\\;dx}=\\frac{\\left(\\log 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[17]:<\/div>\n
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int1<\/span>:<\/span> '<\/span>integrate<\/span>(<\/span>log<\/span>(<\/span>x<\/span>)<\/span>\/<\/span>x<\/span>, x<\/span>)<\/span>;\r\n<\/pre>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n
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Out[17]:<\/div>\n
\\[\\tag{${\\it \\%o}_{18}$}\\int {\\frac{\\log x}{x}}{\\;dx}\\]<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n
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$t = \\log x$ \u3068\u5909\u6570\u5909\u63db\u3092\u3059\u308b\u3002\u5909\u6570\u5909\u63db\u3092\u884c\u3046\u306e\u304c changevar()<\/code> \u95a2\u6570\u3002<\/p>\n<\/div>\n<\/div>\n<\/div>\n
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In\u00a0[18]:<\/div>\n
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eq1<\/span>:<\/span> t<\/span> =<\/span> log<\/span>(<\/span>x<\/span>)<\/span>;\r\nint2<\/span>:<\/span> changevar<\/span>(<\/span>int1<\/span>, eq1<\/span>, t<\/span>, x<\/span>)<\/span>;\r\n<\/pre>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n
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Out[18]:<\/div>\n
\\[\\tag{${\\it \\%o}_{19}$}t=\\log x\\]<\/div>\n<\/div>\n
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Out[18]:<\/div>\n
\\[\\tag{${\\it \\%o}_{20}$}\\int {t}{\\;dt}\\]<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n
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\u5909\u6570\u5909\u63db\u3057\u305f\u5f8c\u306e\u3053\u306e\u7a4d\u5206\u3092\u5b9f\u884c\u3057\u307e\u3059\u3002<\/p>\n
ev()<\/code> \u95a2\u6570\u306f evaluate \u3064\u307e\u308a\u8a55\u4fa1\u3059\u308b\u3068\u3044\u3046\u610f\u5473\u3067\u3059\u3002\u4ee5\u4e0b\u306e\u4f8b\u3067\u306f\uff0cintegrate<\/code> \u3068\u3044\u3046\u30aa\u30d7\u30b7\u30e7\u30f3\u3092\u3064\u3051\u3066\uff0cint2<\/code> \u306e\u7a4d\u5206\u3092\u5b9f\u969b\u306b\u884c\u306a\u3063\u3066\u8a55\u4fa1\u3059\u308b\u3068\u3044\u3046\u3053\u3068\u306b\u306a\u308a\u307e\u3059\u3002<\/p>\n<\/div>\n<\/div>\n<\/div>\n
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In\u00a0[19]:<\/div>\n
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int2<\/span> =<\/span> ev<\/span>(<\/span>int2<\/span>, integrate<\/span>)<\/span>;\r\n<\/pre>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n
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Out[19]:<\/div>\n
\\[\\tag{${\\it \\%o}_{21}$}\\int {t}{\\;dt}=\\frac{t^2}{2}\\]<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n
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\u6700\u5f8c\u306b\uff0c\u76f4\u524d\u306e\u7d50\u679c %<\/code> \u306e\u53f3\u8fba rhs(%)<\/code> \u306b $t = \\log x$ \u3092 subst()<\/code> \u95a2\u6570\u3067\u4ee3\u5165\u3057\u307e\u3059\u3002<\/p>\n<\/div>\n<\/div>\n<\/div>\n
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In\u00a0[20]:<\/div>\n
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ev<\/span>(<\/span>int2<\/span>, integrate<\/span>)<\/span> =<\/span> subst<\/span>(<\/span>eq1<\/span>, rhs<\/span>(<\/span>%<\/span>))<\/span>;\r\n<\/pre>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n
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Out[20]:<\/div>\n
\\[\\tag{${\\it \\%o}_{22}$}\\frac{t^2}{2}=\\frac{\\left(\\log 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[21]:<\/div>\n
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'<\/span>integrate<\/span>(<\/span>log<\/span>(<\/span>x<\/span>)<\/span>, x<\/span>)<\/span> =<\/span> \r\n integrate<\/span>(<\/span>log<\/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[21]:<\/div>\n
\\[\\tag{${\\it \\%o}_{23}$}\\int {\\log x}{\\;dx}=x\\,\\log x-x\\]<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/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[22]:<\/div>\n
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'<\/span>integrate<\/span>(<\/span>asin<\/span>(<\/span>x<\/span>)<\/span>, x<\/span>)<\/span> =<\/span> \r\n integrate<\/span>(<\/span>asin<\/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[22]:<\/div>\n
\\[\\tag{${\\it \\%o}_{24}$}\\int {\\arcsin x}{\\;dx}=x\\,\\arcsin x+\\sqrt{1-x^2}\\]<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n
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In\u00a0[23]:<\/div>\n
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'<\/span>integrate<\/span>(<\/span>acos<\/span>(<\/span>x<\/span>)<\/span>, x<\/span>)<\/span> =<\/span> \r\n integrate<\/span>(<\/span>acos<\/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
\\[\\tag{${\\it \\%o}_{25}$}\\int {\\arccos x}{\\;dx}=x\\,\\arccos x-\\sqrt{1-x^2}\\]<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n
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In\u00a0[24]:<\/div>\n
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'<\/span>integrate<\/span>(<\/span>atan<\/span>(<\/span>x<\/span>)<\/span>, x<\/span>)<\/span> =<\/span> \r\n integrate<\/span>(<\/span>atan<\/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
\\[\\tag{${\\it \\%o}_{26}$}\\int {\\arctan x}{\\;dx}=x\\,\\arctan x-\\frac{\\log \\left(x^2+1\\right)}{2}\\]<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/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[25]:<\/div>\n
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'<\/span>integrate<\/span>(<\/span>asinh<\/span>(<\/span>x<\/span>)<\/span>, x<\/span>)<\/span> =<\/span> \r\n integrate<\/span>(<\/span>asinh<\/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[25]:<\/div>\n
\\[\\tag{${\\it \\%o}_{27}$}\\int {{\\rm asinh}\\; x}{\\;dx}=x\\,{\\rm asinh}\\; x-\\sqrt{x^2+1}\\]<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n
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In\u00a0[26]:<\/div>\n
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'<\/span>integrate<\/span>(<\/span>acosh<\/span>(<\/span>x<\/span>)<\/span>, x<\/span>)<\/span> =<\/span> \r\n 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[26]:<\/div>\n
\\[\\tag{${\\it \\%o}_{28}$}\\int {{\\rm acosh}\\; x}{\\;dx}=x\\,{\\rm acosh}\\; x-\\sqrt{x^2-1}\\]<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n
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In\u00a0[27]:<\/div>\n
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'<\/span>integrate<\/span>(<\/span>atanh<\/span>(<\/span>x<\/span>)<\/span>, x<\/span>)<\/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[27]:<\/div>\n
\\[\\tag{${\\it \\%o}_{29}$}\\int {{\\rm atanh}\\; x}{\\;dx}=\\frac{\\log \\left(1-x^2\\right)}{2}+x\\,{\\rm atanh}\\; x\\]<\/div>\n<\/div>\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
Maxima \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[28]:<\/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\n<\/pre>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n
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Out[28]:<\/div>\n
\\[\\tag{${\\it \\%o}_{30}$}\\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[29]:<\/div>\n
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integrate<\/span>(<\/span>f<\/span>, x<\/span>)<\/span>;\r\n<\/pre>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n
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Out[29]:<\/div>\n
\\[\\tag{${\\it \\%o}_{31}$}\\frac{\\log \\left(x+2\\right)}{3}+\\frac{2\\,x^2+2\\,x}{2}+\\frac{2\\,\\log \\left(x-1\\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\uff0cMaxima \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 partfrac()<\/code> \u95a2\u6570\u3067\u90e8\u5206\u5206\u6570\u306b\u5206\u89e3\u3057\u3066\u307f\u308b\u3068\uff0c\u7a4d\u5206\u306e\u7d50\u679c\u3082\u7d0d\u5f97\u3067\u304d\u307e\u3059\u3002<\/p>\n
\u4f7f\u3044\u65b9\u306f\u4ee5\u4e0b\u306e\u3088\u3046\u306b\u3002<\/p>\n<\/div>\n<\/div>\n<\/div>\n
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In\u00a0[30]:<\/div>\n
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f<\/span> =<\/span> f1<\/span>:<\/span> partfrac<\/span>(<\/span>f<\/span>, x<\/span>)<\/span>;\r\n<\/pre>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n
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Out[30]:<\/div>\n
\\[\\tag{${\\it \\%o}_{32}$}\\frac{2\\,x^3+3\\,x^2-2\\,x-1}{x^2+x-2}=\\frac{1}{3\\,\\left(x+2\\right)}+2\\,x+\\frac{2}{3\\,\\left(x-1\\right)}+1\\]<\/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\\frac{1}{3} \\int \\frac{dx}{x+2} + \\int (2 x+1) \\, dx +
\n\\frac{2}{3}\\int \\frac{dx}{x-1} \\\\
\n&=& \\frac{1}{3} \\log(x+2) + x^2 + x + \\frac{2}{3} \\log(x-1)
\n\\end{eqnarray}<\/p>\n<\/div>\n<\/div>\n<\/div>\n
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In\u00a0[31]:<\/div>\n
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'<\/span>integrate<\/span>(<\/span>f1<\/span>, x<\/span>)<\/span> =<\/span> \r\n integrate<\/span>(<\/span>f1<\/span>, x<\/span>)<\/span>;\r\n<\/pre>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n
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Out[31]:<\/div>\n
\\[\\tag{${\\it \\%o}_{33}$}\\int {\\frac{1}{3\\,\\left(x+2\\right)}+2\\,x+\\frac{2}{3\\,\\left(x-1\\right)}+1}{\\;dx}=\\frac{\\log \\left(x+2\\right)}{3}+x^2+x+\\frac{2\\,\\log \\left(x-1\\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[32]:<\/div>\n
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'<\/span>integrate<\/span>(<\/span>1\/<\/span>cos<\/span>(<\/span>x<\/span>)<\/span>, x<\/span>)<\/span> =<\/span>\r\n integrate<\/span>(<\/span>1\/<\/span>cos<\/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[32]:<\/div>\n
\\[\\tag{${\\it \\%o}_{34}$}\\int {\\frac{1}{\\cos x}}{\\;dx}=\\frac{\\log \\left(\\sin x+1\\right)}{2}-\\frac{\\log \\left(\\sin x-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[33]:<\/div>\n
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int1<\/span>:<\/span> '<\/span>integrate<\/span>(<\/span>1\/<\/span>cos<\/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[33]:<\/div>\n
\\[\\tag{${\\it \\%o}_{35}$}\\int {\\frac{1}{\\cos x}}{\\;dx}\\]<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n
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In\u00a0[34]:<\/div>\n
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eq1<\/span>:<\/span> t<\/span> =<\/span> tan<\/span>(<\/span>x<\/span>\/<\/span>2<\/span>)<\/span>;\r\nint2<\/span>:<\/span> changevar<\/span>(<\/span>int1<\/span>, eq1<\/span>, t<\/span>, x<\/span>)<\/span>;\r\n<\/pre>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n
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solve: using arc-trig functions to get a solution.\r\nSome solutions will be lost.\r\n<\/pre>\n<\/div>\n<\/div>\n
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Out[34]:<\/div>\n
\\[\\tag{${\\it \\%o}_{36}$}t=\\tan \\left(\\frac{x}{2}\\right)\\]<\/div>\n<\/div>\n
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Out[34]:<\/div>\n
\\[\\tag{${\\it \\%o}_{37}$}2\\,\\int {\\frac{1}{\\left(t^2+1\\right)\\,\\cos \\left(2\\,\\arctan t\\right)}}{\\;dt}\\]<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n
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\u88ab\u7a4d\u5206\u95a2\u6570\u3092 trigexpand()<\/code> \u3092\u4f7f\u3063\u3066\u3082\u3046\u5c11\u3057\u6574\u7406\u3057\u3066\u307f\u308b\u3002<\/p>\n<\/div>\n<\/div>\n<\/div>\n
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In\u00a0[35]:<\/div>\n
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int2<\/span>:<\/span> changevar<\/span>(<\/span>int1<\/span>, eq1<\/span>, t<\/span>, x<\/span>)<\/span>, trigexpand<\/span>;\r\n<\/pre>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n
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Out[35]:<\/div>\n
\\[\\tag{${\\it \\%o}_{38}$}-2\\,\\int {\\frac{1}{t^2-1}}{\\;dt}\\]<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n
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In\u00a0[36]:<\/div>\n
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int2<\/span> =<\/span> ev<\/span>(<\/span>int2<\/span>, integrate<\/span>)<\/span>, ratsimp<\/span>;\r\n<\/pre>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n
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Out[36]:<\/div>\n
\\[\\tag{${\\it \\%o}_{39}$}-2\\,\\int {\\frac{1}{t^2-1}}{\\;dt}=\\log \\left(t+1\\right)-\\log \\left(t-1\\right)\\]<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n
\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
\n
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In\u00a0[37]:<\/div>\n
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\n
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subst<\/span>(<\/span>eq1<\/span>, rhs<\/span>(<\/span>%<\/span>))<\/span>;\r\n<\/pre>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n
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Out[37]:<\/div>\n
\\[\\tag{${\\it \\%o}_{40}$}\\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
\n
<\/div>\n
\n
\n
\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
\n
<\/div>\n
\n
\n
\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[38]:<\/div>\n
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\/* a > b \u3068\u4eee\u5b9a *\/<\/span>\r\nassume<\/span>(<\/span>a<\/span> ><\/span> b<\/span>)<\/span>$\r\n\r\nf<\/span>:<\/span> (<\/span>a<\/span>-<\/span>b<\/span>*<\/span>cos<\/span>(<\/span>phi<\/span>))<\/span>\/<\/span>(<\/span>a<\/span>**<\/span>2+<\/span>b<\/span>**<\/span>2-<\/span>2*<\/span>a<\/span>*<\/span>b<\/span>*<\/span>cos<\/span>(<\/span>phi<\/span>))<\/span>;\r\nintegrate<\/span>(<\/span>f<\/span>, phi<\/span>)<\/span>, ratsimp<\/span>;\r\n<\/pre>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n
\n
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\n
<\/div>\n
\n
\u001bXIs b + a zero or nonzero?\r\n\u001b\\Is b + a zero or nonzero?\r\nnonzero;\r\n\r\n<\/pre>\n<\/div>\n<\/div>\n
\n
Out[38]:<\/div>\n
\\[\\tag{${\\it \\%o}_{42}$}\\frac{a-b\\,\\cos \\varphi}{-2\\,a\\,b\\,\\cos \\varphi+b^2+a^2}\\]<\/div>\n<\/div>\n
\n
Out[38]:<\/div>\n
\\[\\tag{${\\it \\%o}_{43}$}-\\frac{\\arctan \\left(\\frac{\\left(b+a\\right)\\,\\sin \\varphi}{\\left(b-a\\right)\\,\\cos \\varphi+b-a}\\right)-\\arctan \\left(\\frac{\\sin \\varphi}{\\cos \\varphi+1}\\right)}{a}\\]<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n
\n
<\/div>\n
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\n
\u7a4d\u5206\u3067\u304d\u3066\u3044\u308b\u3088\u3046\u3060\u304c\uff0c\u30bb\u30aa\u30ea\u30fc\u306b\u5f93\u3063\u3066\u7f6e\u63db\u7a4d\u5206\u3057\u3066\u307f\u308b\u3002<\/p>\n
$\\displaystyle \\tan \\frac{\\phi}{2} \\equiv t$ \u3068\u304a\u3044\u3066…<\/p>\n<\/div>\n<\/div>\n<\/div>\n
\n
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In\u00a0[39]:<\/div>\n
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int1<\/span>:<\/span> '<\/span>integrate<\/span>(<\/span>f<\/span>, phi<\/span>)<\/span>;\r\n<\/pre>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n
\n
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Out[39]:<\/div>\n
\\[\\tag{${\\it \\%o}_{44}$}\\int {\\frac{a-b\\,\\cos \\varphi}{-2\\,a\\,b\\,\\cos \\varphi+b^2+a^2}}{\\;d\\varphi}\\]<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n
\n
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In\u00a0[40]:<\/div>\n
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\n
eq1<\/span>:<\/span> t<\/span> =<\/span> tan<\/span>(<\/span>phi<\/span>\/<\/span>2<\/span>)<\/span>;\r\nint2<\/span>:<\/span> changevar<\/span>(<\/span>int1<\/span>, eq1<\/span>, t<\/span>, phi<\/span>)<\/span>, trigexpand<\/span>;\r\n<\/pre>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n
\n
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Out[40]:<\/div>\n
\\[\\tag{${\\it \\%o}_{45}$}t=\\tan \\left(\\frac{\\varphi}{2}\\right)\\]<\/div>\n<\/div>\n
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Out[40]:<\/div>\n
\\[\\tag{${\\it \\%o}_{46}$}\\int {\\frac{\\left(2\\,b+2\\,a\\right)\\,t^2-2\\,b+2\\,a}{\\left(b^2+2\\,a\\,b+a^2\\right)\\,t^4+\\left(2\\,b^2+2\\,a^2\\right)\\,t^2+b^2-2\\,a\\,b+a^2}}{\\;dt}\\]<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n
\n
<\/div>\n
\n
\n
\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[41]:<\/div>\n
\n
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ft<\/span>:<\/span> diff<\/span>(<\/span>int2<\/span>, t<\/span>)<\/span>$\r\nft<\/span>:<\/span> partfrac<\/span>(<\/span>ft<\/span>, t<\/span>)<\/span>;\r\n<\/pre>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n
\n
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Out[41]:<\/div>\n
\\[\\tag{${\\it \\%o}_{48}$}\\frac{1}{a\\,\\left(t^2+1\\right)}-\\frac{b^2-a^2}{a\\,\\left(\\left(b^2+2\\,a\\,b+a^2\\right)\\,t^2+b^2-2\\,a\\,b+a^2\\right)}\\]<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n
\n
<\/div>\n
\n
\n
\u3053\u306e\u5f62\u306e\u88ab\u7a4d\u5206\u95a2\u6570\u3092\u3042\u3089\u305f\u3081\u3066\u7a4d\u5206\u3059\u308b\u3068…<\/p>\n<\/div>\n<\/div>\n<\/div>\n
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In\u00a0[42]:<\/div>\n
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ans<\/span>:<\/span> integrate<\/span>(<\/span>ft<\/span>, t<\/span>)<\/span>, ratsimp<\/span>;\r\n<\/pre>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n
\n
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\n
<\/div>\n
\n
\u001bXIs b + a zero or nonzero?\r\n\u001b\\Is b + a zero or nonzero?\r\nnonzero;\r\n\r\n<\/pre>\n<\/div>\n<\/div>\n
\n
Out[42]:<\/div>\n
\\[\\tag{${\\it \\%o}_{49}$}-\\frac{\\arctan \\left(\\frac{\\left(b+a\\right)\\,t}{b-a}\\right)-\\arctan t}{a}\\]<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n
\n
<\/div>\n
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\u304b\u306a\u308a\u3059\u3063\u304d\u308a\u3057\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[43]:<\/div>\n
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subst<\/span>(<\/span>eq1<\/span>, ans<\/span>)<\/span>, ratsimp<\/span>;\r\n<\/pre>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n
\n
\n
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Out[43]:<\/div>\n
\\[\\tag{${\\it \\%o}_{50}$}-\\frac{\\arctan \\left(\\frac{\\left(b+a\\right)\\,\\tan \\left(\\frac{\\varphi}{2}\\right)}{b-a}\\right)-\\arctan \\tan \\left(\\frac{\\varphi}{2}\\right)}{a}\\]<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n
\n
<\/div>\n
\n
\n
\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
integrate()<\/code> \u3067\u7a4d\u5206\u3067\u304d\u308b\u3002<\/p>\n<\/div>\n<\/div>\n<\/div>\n
\n
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In\u00a0[44]:<\/div>\n
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f<\/span>:<\/span> (<\/span>a<\/span>-<\/span>b<\/span>*<\/span>cos<\/span>(<\/span>theta<\/span>))<\/span>\/<\/span>sqrt<\/span>(<\/span>a<\/span>**<\/span>2+<\/span>b<\/span>**<\/span>2-<\/span>2*<\/span>a<\/span>*<\/span>b<\/span>*<\/span>cos<\/span>(<\/span>theta<\/span>))<\/span>**<\/span>3<\/span> *<\/span> sin<\/span>(<\/span>theta<\/span>)<\/span>;\r\nintegrate<\/span>(<\/span>f<\/span>, theta<\/span>)<\/span>, factor<\/span>;\r\n<\/pre>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n
\n
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Out[44]:<\/div>\n
\\[\\tag{${\\it \\%o}_{51}$}\\frac{\\left(a-b\\,\\cos \\vartheta\\right)\\,\\sin \\vartheta}{\\left(-2\\,a\\,b\\,\\cos \\vartheta+b^2+a^2\\right)^{\\frac{3}{2}}}\\]<\/div>\n<\/div>\n
\n
Out[44]:<\/div>\n
\\[\\tag{${\\it \\%o}_{52}$}-\\frac{a\\,\\cos \\vartheta-b}{a^2\\,\\sqrt{-2\\,a\\,b\\,\\cos \\vartheta+b^2+a^2}}\\]<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n
\n
<\/div>\n
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\n
\u7121\u7406\u95a2\u6570\u306e\u7a4d\u5206<\/h3>\n<\/div>\n<\/div>\n<\/div>\n
\n
<\/div>\n
\n
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\u4f8b 1<\/h4>\n
$\\displaystyle \\int \\frac{dx}{x \\sqrt{x+1}}$<\/p>\n
Maxima \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[45]:<\/div>\n
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'<\/span>integrate<\/span>(<\/span>1\/<\/span>(<\/span>x<\/span> *<\/span> sqrt<\/span>(<\/span>x<\/span>+<\/span>1<\/span>))<\/span>, x<\/span>)<\/span> =<\/span>\r\n integrate<\/span>(<\/span>1\/<\/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
\n
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Out[45]:<\/div>\n
\\[\\tag{${\\it \\%o}_{53}$}\\int {\\frac{1}{x\\,\\sqrt{x+1}}}{\\;dx}=\\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
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\n
\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[46]:<\/div>\n
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int1<\/span>:<\/span> '<\/span>integrate<\/span>(<\/span>1\/<\/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
\n
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Out[46]:<\/div>\n
\\[\\tag{${\\it \\%o}_{54}$}\\int {\\frac{1}{x\\,\\sqrt{x+1}}}{\\;dx}\\]<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n
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In\u00a0[47]:<\/div>\n
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assume<\/span>(<\/span>t<\/span> ><\/span> 0<\/span>)<\/span>$\r\neq1<\/span>:<\/span> t<\/span> =<\/span> sqrt<\/span>(<\/span>x<\/span>+<\/span>1<\/span>)<\/span>;\r\nint2<\/span>:<\/span> changevar<\/span>(<\/span>int1<\/span>, eq1<\/span>, t<\/span>, x<\/span>)<\/span>;\r\n<\/pre>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n
\n
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Out[47]:<\/div>\n
\\[\\tag{${\\it \\%o}_{56}$}t=\\sqrt{x+1}\\]<\/div>\n<\/div>\n
\n
Out[47]:<\/div>\n
\\[\\tag{${\\it \\%o}_{57}$}2\\,\\int {\\frac{1}{t^2-1}}{\\;dt}\\]<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n
\n
<\/div>\n
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\n
\u88ab\u7a4d\u5206\u95a2\u6570\u3092\u90e8\u5206\u5206\u6570\u306b\u5206\u89e3\u3057\u3066…<\/p>\n<\/div>\n<\/div>\n<\/div>\n
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In\u00a0[48]:<\/div>\n
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ft<\/span>:<\/span> diff<\/span>(<\/span>int2<\/span>, t<\/span>)<\/span>$\r\nft<\/span>:<\/span> partfrac<\/span>(<\/span>ft<\/span>, t<\/span>)<\/span>;\r\n<\/pre>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n
\n
\n
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Out[48]:<\/div>\n
\\[\\tag{${\\it \\%o}_{59}$}\\frac{1}{t-1}-\\frac{1}{t+1}\\]<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n
\n
<\/div>\n
\n
\n
\u3053\u308c\u306f $t$ \u3067\u7c21\u5358\u306b\u7a4d\u5206\u3067\u304d\u3066…<\/p>\n<\/div>\n<\/div>\n<\/div>\n
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In\u00a0[49]:<\/div>\n
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ans<\/span>:<\/span> integrate<\/span>(<\/span>ft<\/span>, t<\/span>)<\/span>;\r\n<\/pre>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n
\n
\n
\n
Out[49]:<\/div>\n
\\[\\tag{${\\it \\%o}_{60}$}\\log \\left(t-1\\right)-\\log \\left(t+1\\right)\\]<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n
\n
<\/div>\n
\n
\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[50]:<\/div>\n
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subst<\/span>(<\/span>eq1<\/span>, ans<\/span>)<\/span>;\r\n<\/pre>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n
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Out[50]:<\/div>\n
\\[\\tag{${\\it \\%o}_{61}$}\\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
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\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
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In\u00a0[51]:<\/div>\n
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int1<\/span>:<\/span> '<\/span>integrate<\/span>(<\/span>1\/<\/span>sqrt<\/span>(<\/span>x<\/span>**<\/span>2<\/span> +<\/span> 1<\/span>)<\/span>, x<\/span>)<\/span>$\r\nint1<\/span> =<\/span> ev<\/span>(<\/span>int1<\/span>, integrate<\/span>)<\/span>;\r\n<\/pre>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n
\n
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Out[51]:<\/div>\n
\\[\\tag{${\\it \\%o}_{63}$}\\int {\\frac{1}{\\sqrt{x^2+1}}}{\\;dx}={\\rm asinh}\\; x\\]<\/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[52]:<\/div>\n
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int1<\/span> =<\/span> ev<\/span>(<\/span>int1<\/span>, integrate<\/span>)<\/span>, logarc<\/span>;\r\n<\/pre>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n
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Out[52]:<\/div>\n
\\[\\tag{${\\it \\%o}_{64}$}\\int {\\frac{1}{\\sqrt{x^2+1}}}{\\;dx}=\\log \\left(\\sqrt{x^2+1}+x\\right)\\]<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n
\n
<\/div>\n
\n
\n
\u3042\u3048\u3066\u3053\u308c\u3092\u7f6e\u63db\u7a4d\u5206\u3057\u3066\u307f\u307e\u3059\u3002<\/p>\n
$x + \\sqrt{x^2 + 1} \\equiv t\\ (>0)$ \u3068\u304a\u304f\u3068…<\/p>\n<\/div>\n<\/div>\n<\/div>\n
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In\u00a0[53]:<\/div>\n
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eq1<\/span>:<\/span> t<\/span> =<\/span> x<\/span> +<\/span> sqrt<\/span>(<\/span>x<\/span>**<\/span>2<\/span> +<\/span> 1<\/span>)<\/span>$\r\nint2<\/span>:<\/span> changevar<\/span>(<\/span>int1<\/span>, eq1<\/span>, t<\/span>, x<\/span>)<\/span>;\r\n<\/pre>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n
\n
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Out[53]:<\/div>\n
\\[\\tag{${\\it \\%o}_{66}$}\\int {\\frac{1}{\\sqrt{x^2+1}}}{\\;dx}\\]<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n
\n
<\/div>\n
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\n
… \u3063\u3066\uff0c\u3046\u307e\u304f\u7f6e\u63db\u3057\u3066\u304f\u308c\u307e\u305b\u3093\u306d\u3002Maxima \u306f $t = x + \\sqrt{x^2+1}$ \u3092 $x$ \u306b\u3064\u3044\u3066\u89e3\u304f\u3053\u3068\u304c\u4e0d\u5f97\u624b\u3067\u3059\u3002\u305d\u306e\u3042\u305f\u308a\u304c\u95a2\u4fc2\u3057\u3066\u3044\u308b\u304b\u3082\u77e5\u308c\u307e\u305b\u3093\u3002<\/p>\n<\/div>\n<\/div>\n<\/div>\n
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In\u00a0[54]:<\/div>\n
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solve<\/span>(<\/span>t<\/span> =<\/span> x<\/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[54]:<\/div>\n
\\[\\tag{${\\it \\%o}_{67}$}\\left[ x=t-\\sqrt{x^2+1} \\right] \\]<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n
\n
<\/div>\n
\n
\n
\u4ee5\u4e0b\u306e\u3088\u3046\u306b\u5909\u5f62\u3057\u3066\u304b\u3089 $t$ \u306b\u3064\u3044\u3066\u89e3\u304f\u3068\u7c21\u5358\u306b\u89e3\u304f\u3053\u3068\u304c\u3067\u304d\u307e\u3059\u3002<\/p>\n
\\begin{eqnarray}
\nt &=& x + \\sqrt{x^2 + 1} \\\\
\nt – x &=& \\sqrt{x^2 + 1} \\\\
\n\\therefore\\ \\ (t – x)^2 &=& x^2 + 1 \\\\
\nt^2 – 2 t x + x^2 &=& x^2 + 1 \\\\
\n\\therefore\\ \\ x &=& \\frac{t^2-1}{2t}
\n\\end{eqnarray}<\/p>\n<\/div>\n<\/div>\n<\/div>\n
\n
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In\u00a0[55]:<\/div>\n
\n
\n
\n
eq2<\/span>:<\/span> (<\/span>t<\/span> -<\/span> x<\/span>)<\/span>**<\/span>2<\/span> =<\/span> x<\/span>**<\/span>2<\/span> +<\/span> 1<\/span>;\r\nsolve<\/span>(<\/span>eq2<\/span>, x<\/span>)<\/span>;\r\n<\/pre>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n
\n
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Out[55]:<\/div>\n
\\[\\tag{${\\it \\%o}_{68}$}\\left(t-x\\right)^2=x^2+1\\]<\/div>\n<\/div>\n
\n
Out[55]:<\/div>\n
\\[\\tag{${\\it \\%o}_{69}$}\\left[ x=\\frac{t^2-1}{2\\,t} \\right] \\]<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n
\n
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In\u00a0[56]:<\/div>\n
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chikan2<\/span>:<\/span> %<\/span>[<\/span>1]<\/span>;\r\n<\/pre>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n
\n
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Out[56]:<\/div>\n
\\[\\tag{${\\it \\%o}_{70}$}x=\\frac{t^2-1}{2\\,t}\\]<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n
\n
<\/div>\n
\n
\n
\u3053\u306e\u5909\u6570\u5909\u63db\u306e\u5f0f\u3092\u4f7f\u3063\u3066\u7f6e\u63db\u7a4d\u5206\u3057\u3066\u307f\u307e\u3059\u3002<\/p>\n<\/div>\n<\/div>\n<\/div>\n
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In\u00a0[57]:<\/div>\n
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int2<\/span>:<\/span> changevar<\/span>(<\/span>int1<\/span>, chikan2<\/span>, t<\/span>, x<\/span>)<\/span>;\r\n<\/pre>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n
\n
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Out[57]:<\/div>\n
\\[\\tag{${\\it \\%o}_{71}$}\\int {\\frac{1}{t}}{\\;dt}\\]<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n
\n
<\/div>\n
\n
\n
\u88ab\u7a4d\u5206\u95a2\u6570\u304c\u304d\u308f\u3081\u3066\u7c21\u5358\u306b\u306a\u308a\u307e\u3057\u305f\u3002\u3042\u3068\u306f\u5b9f\u969b\u306b\u7a4d\u5206\u3057\u3066…<\/p>\n<\/div>\n<\/div>\n<\/div>\n
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In\u00a0[58]:<\/div>\n
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ans<\/span>:<\/span> ev<\/span>(<\/span>int2<\/span>, integrate<\/span>)<\/span>;\r\n<\/pre>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n
\n
\n
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Out[58]:<\/div>\n
\\[\\tag{${\\it \\%o}_{72}$}\\log t\\]<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n
\n
<\/div>\n
\n
\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[59]:<\/div>\n
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subst<\/span>(<\/span>eq1<\/span>, ans<\/span>)<\/span>;\r\n<\/pre>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n
\n
\n
\n
Out[59]:<\/div>\n
\\[\\tag{${\\it \\%o}_{73}$}\\log \\left(\\sqrt{x^2+1}+x\\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
\n
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In\u00a0[60]:<\/div>\n
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\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
\n
\n
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Out[60]:<\/div>\n
\\[\\tag{${\\it \\%o}_{74}$}\\frac{{\\rm asinh}\\; x}{2}+\\frac{x\\,\\sqrt{x^2+1}}{2}\\]<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n
\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[61]:<\/div>\n
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logarc<\/span>(<\/span>%<\/span>)<\/span>;\r\n<\/pre>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n
\n
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Out[61]:<\/div>\n
\\[\\tag{${\\it \\%o}_{75}$}\\frac{\\log \\left(\\sqrt{x^2+1}+x\\right)}{2}+\\frac{x\\,\\sqrt{x^2+1}}{2}\\]<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n
\n
<\/div>\n
\n
\n
\u3053\u308c\u3082\uff0c\u30d2\u30f3\u30c8\u306b\u3057\u305f\u304c\u3063\u3066\uff0c\u3042\u3048\u3066\u7f6e\u63db\u7a4d\u5206\u3067\u3084\u3063\u3066\u307f\u308b\u3002<\/p>\n
$x+\\sqrt{x^2+1} \\equiv t$ \u3068\u304a\u304f\u3068\u3053\u308d\u3067\u3059\u304c\uff0c\u4f8b 2 \u3067\u308f\u304b\u3063\u305f\u3088\u3046\u306b\uff0c\u3053\u308c\u3092 $x$ \u306b\u3064\u3044\u3066\u89e3\u3044\u3066<\/p>\n
$$x = \\frac{t^2-1}{2t}$$<\/p>\n
\u3068\u3057\u3066\u7f6e\u63db\u7a4d\u5206\u3057\u307e\u3059\u3002<\/p>\n<\/div>\n<\/div>\n<\/div>\n
\n
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In\u00a0[62]:<\/div>\n
\n
\n
\n
int1<\/span>:<\/span> '<\/span>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
\n
\n
\n
Out[62]:<\/div>\n
\\[\\tag{${\\it \\%o}_{76}$}\\int {\\sqrt{x^2+1}}{\\;dx}\\]<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n
\n
\n
In\u00a0[63]:<\/div>\n
\n
\n
\n
chikan<\/span>:<\/span> x<\/span> =<\/span> (<\/span>t<\/span>**<\/span>2-<\/span>1<\/span>)<\/span>\/<\/span>(<\/span>2*<\/span>t<\/span>)<\/span>;\r\nint2<\/span>:<\/span> changevar<\/span>(<\/span>int1<\/span>, chikan<\/span>, t<\/span>, x<\/span>)<\/span>, expand<\/span>;\r\n<\/pre>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n
\n
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Out[63]:<\/div>\n
\\[\\tag{${\\it \\%o}_{77}$}x=\\frac{t^2-1}{2\\,t}\\]<\/div>\n<\/div>\n
\n
Out[63]:<\/div>\n
\\[\\tag{${\\it \\%o}_{78}$}\\int {\\frac{t}{4}+\\frac{1}{2\\,t}+\\frac{1}{4\\,t^3}}{\\;dt}\\]<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n
\n
<\/div>\n
\n
\n
\u88ab\u7a4d\u5206\u95a2\u6570\u304c\u7c21\u5358\u306b\u306a\u3063\u305f\u3068\u3053\u308d\u3067\uff0c$t$ \u3067\u7a4d\u5206\u3057\u307e\u3059\u3002<\/p>\n<\/div>\n<\/div>\n<\/div>\n
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In\u00a0[64]:<\/div>\n
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\n
\n
ans<\/span>:<\/span> ev<\/span>(<\/span>int2<\/span>, integrate<\/span>)<\/span>;\r\n<\/pre>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n
\n
\n
\n
Out[64]:<\/div>\n
\\[\\tag{${\\it \\%o}_{79}$}\\frac{\\log t}{2}+\\frac{t^2}{8}-\\frac{1}{8\\,t^2}\\]<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n
\n
<\/div>\n
\n
\n
$t$ \u3092\u3082\u3068\u306e\u5909\u6570 $x+\\sqrt{x^2+1}$ \u306b\u623b\u3057\u3066…<\/p>\n<\/div>\n<\/div>\n<\/div>\n
\n
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In\u00a0[65]:<\/div>\n
\n
\n
\n
subst<\/span>(<\/span>t<\/span>=<\/span>x<\/span>+<\/span>sqrt<\/span>(<\/span>x<\/span>**<\/span>2+<\/span>1<\/span>)<\/span>, ans<\/span>)<\/span>;\r\n<\/pre>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n
\n
\n
\n
Out[65]:<\/div>\n
\\[\\tag{${\\it \\%o}_{80}$}\\frac{\\log \\left(\\sqrt{x^2+1}+x\\right)}{2}+\\frac{\\left(\\sqrt{x^2+1}+x\\right)^2}{8}-\\frac{1}{8\\,\\left(\\sqrt{x^2+1}+x\\right)^2}\\]<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n
\n
<\/div>\n
\n
\n
Maxima \u3067\u306f\u3053\u308c\u4ee5\u4e0a\u306f\u3084\u3063\u3066\u304f\u308c\u306a\u3044\u3088\u3046\u3067\u3059\u304c\uff0c\u4ee5\u4e0b\u306e\u95a2\u4fc2\u3092\u4f7f\u3046\u3068\u3082\u3063\u3068\u30b3\u30f3\u30d1\u30af\u30c8\u306b\u306a\u308a\u307e\u3059\u3002<\/p>\n
\\begin{eqnarray}
\n\\frac{1}{t} &=&= \\frac{1}{x + \\sqrt{x^2+1}} \\\\
\n&=& \\frac{\\sqrt{x^2+1}-x}{\\left(\\sqrt{x^2+1}+x\\right)\\left(\\sqrt{x^2+1}-x\\right)} \\\\
\n&=& \\frac{\\sqrt{x^2+1}-x}{x^2+1-x^2} \\\\
\n&=& \\sqrt{x^2+1}-x
\n\\end{eqnarray}<\/p>\n<\/div>\n<\/div>\n<\/div>\n
\n
\n
In\u00a0[66]:<\/div>\n
\n
\n
\n
1\/<\/span>8<\/span> *<\/span> ((<\/span>sqrt<\/span>(<\/span>x<\/span>**<\/span>2+<\/span>1<\/span>)<\/span>+<\/span>x<\/span>)<\/span>**<\/span>2<\/span> -<\/span> (<\/span>sqrt<\/span>(<\/span>x<\/span>**<\/span>2+<\/span>1<\/span>)<\/span>-<\/span>x<\/span>)<\/span>**<\/span>2<\/span>)<\/span>, ratsimp<\/span>;\r\n<\/pre>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n
\n
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Out[66]:<\/div>\n
\\[\\tag{${\\it \\%o}_{81}$}\\frac{x\\,\\sqrt{x^2+1}}{2}\\]<\/div>\n<\/div>\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
\n
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In\u00a0[67]:<\/div>\n
\n
\n
\n
integrate<\/span>(<\/span>1\/<\/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
\n
\n
\n
Out[67]:<\/div>\n
\\[\\tag{${\\it \\%o}_{82}$}2\\,\\arctan \\sqrt{x-1}\\]<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n
\n
<\/div>\n
\n
\n
\u3053\u308c\u3092\u3042\u3048\u3066\u7f6e\u63db\u7a4d\u5206\u3067\u3084\u3063\u3066\u307f\u307e\u3059\u3002$t \\equiv \\sqrt{x-1}$ \u3068\u304a\u3044\u3066…<\/p>\n<\/div>\n<\/div>\n<\/div>\n
\n
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In\u00a0[68]:<\/div>\n
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int1<\/span>:<\/span> '<\/span>integrate<\/span>(<\/span>1\/<\/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
\n
\n
\n
Out[68]:<\/div>\n
\\[\\tag{${\\it \\%o}_{83}$}\\int {\\frac{1}{\\sqrt{x-1}\\,x}}{\\;dx}\\]<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n
\n
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In\u00a0[69]:<\/div>\n
\n
\n
\n
eq1<\/span>:<\/span> t<\/span> =<\/span> sqrt<\/span>(<\/span>x<\/span>-<\/span>1<\/span>)<\/span>;\r\nint2<\/span>:<\/span> changevar<\/span>(<\/span>int1<\/span>, eq1<\/span>, t<\/span>, x<\/span>)<\/span>;\r\n<\/pre>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n
\n
\n
\n
Out[69]:<\/div>\n
\\[\\tag{${\\it \\%o}_{84}$}t=\\sqrt{x-1}\\]<\/div>\n<\/div>\n
\n
Out[69]:<\/div>\n
\\[\\tag{${\\it \\%o}_{85}$}2\\,\\int {\\frac{1}{t^2+1}}{\\;dt}\\]<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n
\n
\n
In\u00a0[70]:<\/div>\n
\n
\n
\n
ans<\/span>:<\/span> ev<\/span>(<\/span>int2<\/span>, integrate<\/span>)<\/span>;\r\n<\/pre>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n
\n
\n
\n
Out[70]:<\/div>\n
\\[\\tag{${\\it \\%o}_{86}$}2\\,\\arctan t\\]<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n
\n
<\/div>\n
\n
\n
\u6700\u5f8c\u306b $t$ \u3092\u3082\u3068\u306e\u5909\u6570 $\\sqrt{x-1}$ \u3067\u3042\u3089\u308f\u3057\u3066…<\/p>\n<\/div>\n<\/div>\n<\/div>\n
\n
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In\u00a0[71]:<\/div>\n
\n
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subst<\/span>(<\/span>eq1<\/span>, ans<\/span>)<\/span>;\r\n<\/pre>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n
\n
\n
\n
Out[71]:<\/div>\n
\\[\\tag{${\\it \\%o}_{87}$}2\\,\\arctan \\sqrt{x-1}\\]<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n
\n
<\/div>\n
\n
\n
\u7df4\u7fd2\u554f\u984c 2<\/h4>\n
$\\displaystyle \\int \\frac{\\sqrt{x}}{\\sqrt{1-x}} dx$<\/p>\n<\/div>\n<\/div>\n<\/div>\n
\n
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In\u00a0[72]:<\/div>\n
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\n
\n
integrate<\/span>(<\/span>sqrt<\/span>(<\/span>x<\/span>)<\/span>\/<\/span>sqrt<\/span>(<\/span>x<\/span>-<\/span>1<\/span>)<\/span>, x<\/span>)<\/span>, expand<\/span>;\r\n<\/pre>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n
\n
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Out[72]:<\/div>\n
\\[\\tag{${\\it \\%o}_{88}$}\\sqrt{1-\\frac{1}{x}}\\,x+\\frac{\\log \\left(\\sqrt{1-\\frac{1}{x}}+1\\right)}{2}-\\frac{\\log \\left(\\sqrt{1-\\frac{1}{x}}-1\\right)}{2}\\]<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n
\n
<\/div>\n
\n
\n
Maxima \u3067\u306f integrate()<\/code> \u3059\u308c\u3070\u3059\u3050\u306b\u4e0a\u5f0f\u306e\u3088\u3046\u306b\u7b54\u3048\u304c\u3067\u307e\u3059\u304c\uff0c\u6ce8\u610f\u304c\u5fc5\u8981\u3067\u3059\u3002<\/p>\n
\u984c\u610f\u304b\u3089 $0 < x < 1$ \u3067\u3059\u304c\uff0c\u3053\u306e\u7b54\u3048\u3060\u3068 $\\log$ \u306e\u306a\u304b\u306e\u5e73\u65b9\u6839\u306e\u4e2d\u8eab\u304c\u8ca0\u306b\u306a\u3063\u3066\u3057\u307e\u3044\u307e\u3059\u3002<\/p>\n
\u3053\u3053\u306f\u614e\u91cd\u306b\u7f6e\u63db\u7a4d\u5206\u3057\u3066\u78ba\u8a8d\u3057\u3066\u307f\u307e\u3059\u3002<\/p>\n
$\\displaystyle \\frac{\\sqrt{x}}{\\sqrt{1-x}} \\equiv t$ \u3068\u304a\u3044\u3066… \u3068\u3044\u304d\u305f\u3044\u3068\u3053\u308d\u3067\u3059\u304c\uff0cMaxima \u306e\u5f97\u624b\u4e0d\u5f97\u624b\u3092\u8003\u616e\u3057\u3066\uff0c\u3053\u306e\u5909\u6570\u5909\u63db\u3092 $x$ \u306b\u3064\u3044\u3066\u89e3\u3044\u3066\u304a\u304d\u307e\u3059\u3002<\/p>\n
\\begin{eqnarray}
\nt &=& \\frac{\\sqrt{x}}{\\sqrt{1-x}} \\\\
\n\\sqrt{1-x} t &=& \\sqrt{x}\\\\
\n(1-x) t^2 &=& x \\\\
\n\\therefore\\ \\ x &=& \\frac{t^2}{1+t^2}
\n\\end{eqnarray}<\/p>\n<\/div>\n<\/div>\n<\/div>\n
\n
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In\u00a0[73]:<\/div>\n
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\n
\n
int1<\/span>:<\/span> '<\/span>integrate<\/span>(<\/span>sqrt<\/span>(<\/span>x<\/span>)<\/span>\/<\/span>sqrt<\/span>(<\/span>1-<\/span>x<\/span>)<\/span>, x<\/span>)<\/span>;\r\n<\/pre>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n
\n
\n
\n
Out[73]:<\/div>\n
\\[\\tag{${\\it \\%o}_{89}$}\\int {\\frac{\\sqrt{x}}{\\sqrt{1-x}}}{\\;dx}\\]<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n
\n
\n
In\u00a0[74]:<\/div>\n
\n
\n
\n
int2<\/span>:<\/span> changevar<\/span>(<\/span>int1<\/span>, x<\/span> =<\/span> t<\/span>**<\/span>2\/<\/span>(<\/span>1+<\/span>t<\/span>**<\/span>2<\/span>)<\/span>, t<\/span>, x<\/span>)<\/span>;\r\n<\/pre>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n
\n
\n
\n
Out[74]:<\/div>\n
\\[\\tag{${\\it \\%o}_{90}$}2\\,\\int {\\frac{t^2}{t^4+2\\,t^2+1}}{\\;dt}\\]<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n
\n
\n
In\u00a0[75]:<\/div>\n
\n
\n
\n
ans<\/span>:<\/span> ev<\/span>(<\/span>int2<\/span>, integrate<\/span>)<\/span>, expand<\/span>;\r\n<\/pre>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n
\n
\n
\n
Out[75]:<\/div>\n
\\[\\tag{${\\it \\%o}_{91}$}\\arctan t-\\frac{2\\,t}{2\\,t^2+2}\\]<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n
\n
<\/div>\n
\n
\n
2\u9805\u76ee\u306e\u5206\u6bcd\u306e 2<\/code> \u3092\u7d04\u5206\u3057\u3066\u304f\u308c\u306a\u3044\u306e\u3067\u4eba\u529b\u3067\u3002<\/p>\n<\/div>\n<\/div>\n<\/div>\n
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In\u00a0[76]:<\/div>\n
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\n
\n
ans2<\/span>:<\/span> atan<\/span>(<\/span>t<\/span>)<\/span> +<\/span> factor<\/span>(<\/span>ans<\/span> -<\/span> atan<\/span>(<\/span>t<\/span>))<\/span>;\r\n<\/pre>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n
\n
\n
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Out[76]:<\/div>\n
\\[\\tag{${\\it \\%o}_{92}$}\\arctan t-\\frac{t}{t^2+1}\\]<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n
\n
<\/div>\n
\n
\n
$\\displaystyle x = \\frac{t^2}{t^2+1}$ \u3060\u3063\u305f\u3053\u3068\u3092\u601d\u3044\u51fa\u3057\u3066…<\/p>\n<\/div>\n<\/div>\n<\/div>\n
\n
\n
In\u00a0[77]:<\/div>\n
\n
\n
\n
ans3<\/span>:<\/span> atan<\/span>(<\/span>t<\/span>)<\/span> -<\/span> x<\/span>\/<\/span>t<\/span>;\r\n<\/pre>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n
\n
\n
\n
Out[77]:<\/div>\n
\\[\\tag{${\\it \\%o}_{93}$}\\arctan t-\\frac{x}{t}\\]<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n
\n
<\/div>\n
\n
\n
\u6700\u5f8c\u306b\u6b8b\u308a\u306e $t$ \u3092\u3082\u3068\u306e\u5909\u6570 $\\displaystyle \\frac{\\sqrt{x}}{\\sqrt{1-x}}$ \u3067\u3042\u3089\u308f\u3057\u3066…<\/p>\n<\/div>\n<\/div>\n<\/div>\n
\n
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In\u00a0[78]:<\/div>\n
\n
\n
\n
subst<\/span>(<\/span>t<\/span> =<\/span> sqrt<\/span>(<\/span>x<\/span>)<\/span>\/<\/span>sqrt<\/span>(<\/span>1-<\/span>x<\/span>)<\/span>, ans3<\/span>)<\/span>;\r\n<\/pre>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n
\n
\n
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Out[78]:<\/div>\n
\\[\\tag{${\\it \\%o}_{94}$}\\arctan \\left(\\frac{\\sqrt{x}}{\\sqrt{1-x}}\\right)-\\sqrt{1-x}\\,\\sqrt{x}\\]<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n
\n
<\/div>\n
\n
\n
\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
\n
\n
In\u00a0[79]:<\/div>\n
\n
\n
\n
integrate<\/span>(<\/span>sqrt<\/span>(<\/span>x<\/span>)<\/span>\/<\/span>sqrt<\/span>(<\/span>1-<\/span>x<\/span>**<\/span>3<\/span>)<\/span>, x<\/span>)<\/span>;\r\n<\/pre>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n
\n
\n
\n
Out[79]:<\/div>\n
\\[\\tag{${\\it \\%o}_{95}$}-\\frac{2\\,\\arctan \\left(\\frac{\\sqrt{1-x^3}}{x^{\\frac{3}{2}}}\\right)}{3}\\]<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n
\n
<\/div>\n
\n
\n
Maxima \u306f\u4e0a\u8a18\u306e\u3088\u3046\u306b\u4f55\u98df\u308f\u306c\u9854\u3067 integrate()<\/code> \u3057\u3066\u304f\u308c\u308b\u304c\uff0c\u7b54\u3048\u306b\u8ca0\u53f7\u304c\u3064\u3044\u3066\u3044\u308b\u306e\u304c\u3069\u3046\u3082\u3057\u3063\u304f\u308a\u3053\u306a\u3044\u3002<\/p>\n
\u5b9f\u306f\uff0c\u4ee5\u4e0b\u306e\u95a2\u4fc2<\/p>\n
$$ \\tan^{-1} x + \\tan^{-1} \\frac{1}{x} = \\frac{\\pi}{2}$$<\/p>\n
\u3092\u4f7f\u3046\u3068\uff0c<\/p>\n
$$\\arctan \\left(\\frac{\\sqrt{1-x^3}}{x^{\\frac{3}{2}}}\\right) = – \\arctan \\left(\\frac{x^{\\frac{3}{2}}}{\\sqrt{1-x^3}}\\right) + \\frac{\\pi}{2}$$<\/p>\n
\u3068\u306a\u308b\u306e\u3067\uff0c\u7a4d\u5206\u5b9a\u6570\u306e\u4efb\u610f\u6027\u3092\u9664\u3044\u3066<\/p>\n
$$ \\int \\frac{\\sqrt{x}}{\\sqrt{1-x^3}} dx
\n=
\n\\frac{2}{3} \\arctan \\left(\\frac{\\sqrt{x^3}}{\\sqrt{1-x^3}}\\right) + C$$<\/p>\n
\u3068\u66f8\u3044\u3066\u3082\u3088\u3044\u3002\u8ca0\u53f7\u304c\u3064\u304b\u306a\u3044\u3053\u3063\u3061\u306e\u307b\u3046\u304c\u3057\u3063\u304f\u308a\u304f\u308b\u3002Maxima \u306f\u3053\u3053\u307e\u3067\u306f\u3084\u3063\u3066\u304f\u308c\u306a\u3044\u3002<\/p>\n
\u3061\u306a\u307f\u306b\uff0c$\\displaystyle y = \\arctan \\left(\\frac{\\sqrt{x^3}}{\\sqrt{1-x^3}}\\right)$ \u3068\u304a\u304f\u3068\uff0c<\/p>\n
\\begin{eqnarray}
\n\\frac{\\sqrt{x^3}}{\\sqrt{1-x^3}} &=& \\tan y \\\\
\nx^3 &=& (1-x^3) \\tan^2 y \\\\
\n\\therefore\\ \\ x^3 &=& \\frac{\\tan^2 y}{1 + \\tan^2 y} \\\\
\n&=& \\sin^2 y \\\\
\n\\therefore\\ \\ y &=& \\arcsin x^{\\frac{3}{2}}
\n\\end{eqnarray}<\/p>\n
\u3067\u3042\u308b\u306e\u3067\uff0c<\/p>\n
\\begin{eqnarray}
\n\\int \\frac{\\sqrt{x}}{\\sqrt{1-x^3}} dx &=&
\n\\frac{2}{3} \\arctan \\left(\\frac{\\sqrt{x^3}}{\\sqrt{1-x^3}}\\right) + C\\\\
\n&=& \\frac{2}{3} \\arcsin x^{\\frac{3}{2}} + C
\n\\end{eqnarray}<\/p>\n
\u3068\u3057\u3066\u3082\u3088\u3044\u3002<\/p>\n<\/div>\n<\/div>\n<\/div>\n
\n
<\/div>\n
\n
\n
\u3053\u308c\u3092\u3042\u3048\u3066\u7f6e\u63db\u7a4d\u5206\u3067\u3084\u3063\u3066\u307f\u308b\u3002\u307e\u305a\u306f<\/p>\n
$\\displaystyle t \\equiv \\frac{\\sqrt{x^3}}{\\sqrt{1-x^3}}$ \u3068\u5909\u6570\u5909\u63db\u3059\u308b\u3002Maxima \u306e\u305f\u3081\u306b\u3053\u306e\u5909\u6570\u5909\u63db\u3092 $x$ \u306b\u3064\u3044\u3066\u89e3\u3044\u3066<\/p>\n
\\begin{eqnarray}
\n\\frac{\\sqrt{x^3}}{\\sqrt{1-x^3}} &=& t \\\\
\nx^3 &=& (1 – x^3) t^2 \\\\
\n\\therefore\\ \\ x^3 &=& \\frac{t^2}{1+t^2} \\\\
\nx &=& \\left( \\frac{t^2}{1+t^2}\\right)^{\\frac{1}{3}}
\n\\end{eqnarray}<\/p>\n<\/div>\n<\/div>\n<\/div>\n
\n
\n
In\u00a0[80]:<\/div>\n
\n
\n
\n
int1<\/span>:<\/span> '<\/span>integrate<\/span>(<\/span>sqrt<\/span>(<\/span>x<\/span>)<\/span>\/<\/span>sqrt<\/span>(<\/span>1-<\/span>x<\/span>**<\/span>3<\/span>)<\/span>, x<\/span>)<\/span>;\r\n<\/pre>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n
\n
\n
\n
Out[80]:<\/div>\n
\\[\\tag{${\\it \\%o}_{96}$}\\int {\\frac{\\sqrt{x}}{\\sqrt{1-x^3}}}{\\;dx}\\]<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n
\n
\n
In\u00a0[81]:<\/div>\n
\n
\n
\n
chikan1<\/span>:<\/span> t<\/span> =<\/span> sqrt<\/span>(<\/span>x<\/span>**<\/span>3<\/span>)<\/span>\/<\/span>sqrt<\/span>(<\/span>1-<\/span>x<\/span>**<\/span>3<\/span>)<\/span>;\r\nchikan2<\/span>:<\/span> x<\/span> =<\/span> (<\/span>t<\/span>**<\/span>2\/<\/span>(<\/span>1+<\/span>t<\/span>**<\/span>2<\/span>))<\/span>**<\/span>(<\/span>1\/<\/span>3<\/span>)<\/span>;\r\nint2<\/span>:<\/span> changevar<\/span>(<\/span>int1<\/span>, chikan2<\/span>, t<\/span>, x<\/span>)<\/span>;\r\n<\/pre>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n
\n
\n
\n
Out[81]:<\/div>\n
\\[\\tag{${\\it \\%o}_{97}$}t=\\frac{x^{\\frac{3}{2}}}{\\sqrt{1-x^3}}\\]<\/div>\n<\/div>\n
\n
Out[81]:<\/div>\n
\\[\\tag{${\\it \\%o}_{98}$}x=\\frac{t^{\\frac{2}{3}}}{\\left(t^2+1\\right)^{\\frac{1}{3}}}\\]<\/div>\n<\/div>\n
\n
Out[81]:<\/div>\n
\\[\\tag{${\\it \\%o}_{99}$}2\\,\\int {\\frac{1}{3\\,t^2+3}}{\\;dt}\\]<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n
\n
\n
In\u00a0[82]:<\/div>\n
\n
\n
\n
ans<\/span>:<\/span> ev<\/span>(<\/span>int2<\/span>, integrate<\/span>)<\/span>;\r\n<\/pre>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n
\n
\n
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Out[82]:<\/div>\n
\\[\\tag{${\\it \\%o}_{100}$}\\frac{2\\,\\arctan t}{3}\\]<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n
\n
\n
In\u00a0[83]:<\/div>\n
\n
\n
\n
subst<\/span>(<\/span>chikan1<\/span>, ans<\/span>)<\/span>;\r\n<\/pre>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n
\n
\n
\n
Out[83]:<\/div>\n
\\[\\tag{${\\it \\%o}_{101}$}\\frac{2\\,\\arctan \\left(\\frac{x^{\\frac{3}{2}}}{\\sqrt{1-x^3}}\\right)}{3}\\]<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n
\n
<\/div>\n
\n
\n
\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
\n
\n
In\u00a0[84]:<\/div>\n
\n
\n
\n
integrate<\/span>(<\/span>1\/<\/span>(<\/span>a<\/span>**<\/span>2<\/span> +<\/span> x<\/span>**<\/span>2<\/span>)<\/span>**<\/span>(<\/span>3\/<\/span>2<\/span>)<\/span>, x<\/span>)<\/span>;\r\n<\/pre>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n
\n
\n
\n
Out[84]:<\/div>\n
\\[\\tag{${\\it \\%o}_{102}$}\\frac{x}{a^2\\,\\sqrt{x^2+a^2}}\\]<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n
\n
<\/div>\n
\n
\n
\u5e83\u7fa9\u306e\u7a4d\u5206<\/h3>\n
Maxima \u3067\u306f\u7121\u9650\u5927 $\\infty$ \u306f inf<\/code> \u3067\u3059\u3002<\/p>\n<\/div>\n<\/div>\n<\/div>\n
\n
<\/div>\n
\n
\n
\u7df4\u7fd2\u554f\u984c 1<\/h4>\n
$\\displaystyle \\int_{-\\infty}^{\\infty} \\frac{1}{(a^2 + x^2)^{\\frac{3}{2}}} dx$<\/p>\n
\u4ee5\u4e0b\u3067\u306f $a > 0$ \u3068\u3057\u3066\u8a08\u7b97\u3057\u307e\u3059\u3002<\/p>\n<\/div>\n<\/div>\n<\/div>\n
\n
\n
In\u00a0[85]:<\/div>\n
\n
\n
\n
assume<\/span>(<\/span>a<\/span> ><\/span> 0<\/span>)<\/span>$\r\n'<\/span>integrate<\/span>(<\/span>1\/<\/span>(<\/span>a<\/span>**<\/span>2<\/span> +<\/span> x<\/span>**<\/span>2<\/span>)<\/span>**<\/span>(<\/span>3\/<\/span>2<\/span>)<\/span>, x<\/span>, -<\/span>inf<\/span>, inf<\/span>)<\/span>=<\/span>\r\n integrate<\/span>(<\/span>1\/<\/span>(<\/span>a<\/span>**<\/span>2<\/span> +<\/span> x<\/span>**<\/span>2<\/span>)<\/span>**<\/span>(<\/span>3\/<\/span>2<\/span>)<\/span>, x<\/span>, -<\/span>inf<\/span>, inf<\/span>)<\/span>;\r\n<\/pre>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n
\n
\n
\n
Out[85]:<\/div>\n
\\[\\tag{${\\it \\%o}_{104}$}\\int_{-\\infty }^{\\infty }{\\frac{1}{\\left(x^2+a^2\\right)^{\\frac{3}{2}}}\\;dx}=\\frac{2}{a^2}\\]<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n
\n
<\/div>\n
\n
\n
\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
\n
\n
In\u00a0[86]:<\/div>\n
\n
\n
\n
'<\/span>integrate<\/span>(<\/span>1\/<\/span>(<\/span>a<\/span>**<\/span>2<\/span> +<\/span> x<\/span>**<\/span>2<\/span>)<\/span>, x<\/span>, -<\/span>inf<\/span>, inf<\/span>)<\/span> =<\/span>\r\n integrate<\/span>(<\/span>1\/<\/span>(<\/span>a<\/span>**<\/span>2<\/span> +<\/span> x<\/span>**<\/span>2<\/span>)<\/span>, x<\/span>, -<\/span>inf<\/span>, inf<\/span>)<\/span>;\r\n<\/pre>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n
\n
\n
\n
Out[86]:<\/div>\n
\\[\\tag{${\\it \\%o}_{105}$}\\int_{-\\infty }^{\\infty }{\\frac{1}{x^2+a^2}\\;dx}=\\frac{\\pi}{a}\\]<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n
\n
<\/div>\n
\n
\n
\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
\n
<\/div>\n
\n
\n
\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
\n
\n
In\u00a0[87]:<\/div>\n
\n
\n
\n
assume<\/span>(<\/span>r<\/span> ><\/span> 0<\/span>)<\/span>$\r\n\r\ny<\/span>:<\/span> sqrt<\/span>(<\/span>r<\/span>**<\/span>2<\/span> -<\/span> x<\/span>**<\/span>2<\/span>)<\/span>;\r\n2<\/span> *<\/span> '<\/span>integrate<\/span>(<\/span>y<\/span>, x<\/span>, -<\/span>r<\/span>, r<\/span>)<\/span> =<\/span> \r\n 2<\/span> *<\/span> integrate<\/span>(<\/span>y<\/span>, x<\/span>, -<\/span>r<\/span>, r<\/span>)<\/span>;\r\n<\/pre>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n
\n
\n
\n
Out[87]:<\/div>\n
\\[\\tag{${\\it \\%o}_{107}$}\\sqrt{r^2-x^2}\\]<\/div>\n<\/div>\n
\n
Out[87]:<\/div>\n
\\[\\tag{${\\it \\%o}_{108}$}2\\,\\int_{-r}^{r}{\\sqrt{r^2-x^2}\\;dx}=\\pi\\,r^2\\]<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n
\n
<\/div>\n
\n
\n
\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
\n
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In\u00a0[88]:<\/div>\n
\n
\n
\n
dydx<\/span>:<\/span> diff<\/span>(<\/span>y<\/span>, x<\/span>)<\/span>;\r\n<\/pre>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n
\n
\n
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Out[88]:<\/div>\n
\\[\\tag{${\\it \\%o}_{109}$}-\\frac{x}{\\sqrt{r^2-x^2}}\\]<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n
\n
\n
In\u00a0[89]:<\/div>\n
\n
\n
\n
2*'<\/span>integrate<\/span>(<\/span>sqrt<\/span>(<\/span>1<\/span> +<\/span> dydx<\/span>**<\/span>2<\/span>)<\/span>, x<\/span>, -<\/span>r<\/span>, r<\/span>)<\/span> =<\/span>\r\n 2*<\/span>integrate<\/span>(<\/span>sqrt<\/span>(<\/span>1<\/span> +<\/span> dydx<\/span>**<\/span>2<\/span>)<\/span>, x<\/span>, -<\/span>r<\/span>, r<\/span>)<\/span>;\r\n<\/pre>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n
\n
\n
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Out[89]:<\/div>\n
\\[\\tag{${\\it \\%o}_{110}$}2\\,\\int_{-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
\n
<\/div>\n
\n
\n
\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<\/div>\n<\/div>\n<\/div>\n
\n
\n
In\u00a0[90]:<\/div>\n
\n
\n
\n
'<\/span>integrate<\/span>(<\/span>2*<\/span>%pi<\/span>*<\/span>y<\/span>*<\/span>sqrt<\/span>(<\/span>1+<\/span>dydx<\/span>**<\/span>2<\/span>)<\/span>, x<\/span>, -<\/span>r<\/span>, r<\/span>)<\/span> =<\/span>\r\n integrate<\/span>(<\/span>2*<\/span>%pi<\/span>*<\/span>y<\/span>*<\/span>sqrt<\/span>(<\/span>1+<\/span>dydx<\/span>**<\/span>2<\/span>)<\/span>, x<\/span>, -<\/span>r<\/span>, r<\/span>)<\/span>;\r\n<\/pre>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n
\n
\n
\n
Out[90]:<\/div>\n
\\[\\tag{${\\it \\%o}_{111}$}2\\,\\pi\\,\\int_{-r}^{r}{\\sqrt{r^2-x^2}\\,\\sqrt{\\frac{x^2}{r^2-x^2}+1}\\;dx}=4\\,\\pi\\,r^2\\]<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n
\n
<\/div>\n
\n
\n
\u7403\u306e\u4f53\u7a4d<\/h4>\n
$\\displaystyle V = \\int_{-r}^r \\pi y^2\\, dx$<\/p>\n<\/div>\n<\/div>\n<\/div>\n
\n
\n
In\u00a0[91]:<\/div>\n
\n
\n
\n
'<\/span>integrate<\/span>(<\/span>%pi<\/span>*<\/span>y<\/span>**<\/span>2<\/span>, x<\/span>, -<\/span>r<\/span>, r<\/span>)<\/span> =<\/span>\r\n integrate<\/span>(<\/span>%pi<\/span>*<\/span>y<\/span>**<\/span>2<\/span>, x<\/span>, -<\/span>r<\/span>, r<\/span>)<\/span>;\r\n<\/pre>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n
\n
\n
\n
Out[91]:<\/div>\n
\\[\\tag{${\\it \\%o}_{112}$}\\pi\\,\\int_{-r}^{r}{r^2-x^2\\;dx}=\\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":6032,"menu_order":30,"comment_status":"closed","ping_status":"closed","template":"","meta":{"inline_featured_image":false,"footnotes":""},"class_list":["post-2197","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\/2197","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=2197"}],"version-history":[{"count":6,"href":"https:\/\/home.hirosaki-u.ac.jp\/relativity\/wp-json\/wp\/v2\/pages\/2197\/revisions"}],"predecessor-version":[{"id":6099,"href":"https:\/\/home.hirosaki-u.ac.jp\/relativity\/wp-json\/wp\/v2\/pages\/2197\/revisions\/6099"}],"up":[{"embeddable":true,"href":"https:\/\/home.hirosaki-u.ac.jp\/relativity\/wp-json\/wp\/v2\/pages\/6032"}],"wp:attachment":[{"href":"https:\/\/home.hirosaki-u.ac.jp\/relativity\/wp-json\/wp\/v2\/media?parent=2197"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}