{"id":9512,"date":"2024-11-13T22:00:58","date_gmt":"2024-11-13T13:00:58","guid":{"rendered":"https:\/\/home.hirosaki-u.ac.jp\/relativity\/?page_id=9512"},"modified":"2024-11-14T10:08:37","modified_gmt":"2024-11-14T01:08:37","slug":"%e5%be%ae%e5%88%86%e3%83%bb%e7%a9%8d%e5%88%86","status":"publish","type":"page","link":"https:\/\/home.hirosaki-u.ac.jp\/relativity\/%e3%82%b3%e3%83%b3%e3%83%94%e3%83%a5%e3%83%bc%e3%82%bf%e6%bc%94%e7%bf%92\/sympy-%e6%bc%94%e7%bf%92\/%e5%be%ae%e5%88%86%e3%83%bb%e7%a9%8d%e5%88%86\/","title":{"rendered":"3. SymPy \u3067\u5fae\u5206\u30fb\u7a4d\u5206"},"content":{"rendered":"
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1\u5909\u6570\u95a2\u6570\uff0c\u7279\u306b\uff08\u53cc\u66f2\u7dda\u95a2\u6570\u30fb\u9006\u53cc\u66f2\u7dda\u95a2\u6570\u3092\u542b\u3080\uff09\u521d\u7b49\u95a2\u6570\u306e\u5fae\u5206\u30fb\u7a4d\u5206\u3002\u307e\u305f\uff0c2\u5909\u6570\u95a2\u6570\u306b\u95a2\u3059\u308b\u504f\u5fae\u5206\u30682\u91cd\u8cac\u5206\uff0c\u304a\u3088\u30732\u91cd\u7a4d\u5206\u306b\u95a2\u9023\u3057\u305f\u30ac\u30a6\u30b9\u7a4d\u5206\u306b\u3064\u3044\u3066\u3002<\/p>\n<\/div>\n<\/div>\n<\/div>\n

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SymPy \u306e import<\/h3>\n<\/div>\n<\/div>\n<\/div>\n
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In\u00a0[1]:<\/div>\n
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from<\/span> sympy.abc<\/span> import<\/span> *<\/span>\r\nfrom<\/span> sympy<\/span> import<\/span> *<\/span>\r\n<\/pre>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n
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1\u5909\u6570\u95a2\u6570\u306e\u5fae\u5206 diff()<\/code><\/h3>\n
1\u968e\u5fae\u5206<\/h4>\n
SymPy \u30671\u5909\u6570\u95a2\u6570 $y = f(x)$ \u306e\u5fae\u5206\u306b\u306f\u4ee5\u4e0b\u306e\u3088\u3046\u306b diff()<\/code> \u3092\u4f7f\u3044\u307e\u3059\u3002<\/p>\n
$\\displaystyle \\frac{df}{dx} = \\frac{d}{dx} f(x) =$ diff(f(x), x)<\/code> \uff08$=$ Derivative(f(x), x).doit()<\/code>\uff09<\/p>\n
\u9ad8\u968e\u5fae\u5206<\/h4>\n
2\u968e\u4ee5\u4e0a\u306e\u9ad8\u968e\u5fae\u5206\u306f<\/p>\n
$\\displaystyle \\frac{d^nf}{dx^n} = \\frac{d^n}{dx^n} f(x) =$ diff(f(x), x, n)<\/code><\/p>\n
\u53c2\u8003\uff1aDerivative()<\/code> \u3068 diff()<\/code><\/h4>\n
Derivative()<\/code> \u306f\u5fae\u5206\u306e\u8868\u793a\u3092\u3059\u308b\u3060\u3051\u3067\u5b9f\u969b\u306b\u5fae\u5206\u306f\u884c\u3044\u307e\u305b\u3093\u3002\u5fae\u5206\u3092\u5b9f\u884c\u3059\u308b\u306b\u306f\uff0cDerivative().doit()<\/code> \u306e\u3088\u3046\u306b .doit()<\/code> \u3092\u8ffd\u52a0\u3057\u307e\u3059\u3002<\/p>\n
diff()<\/code> \u3092\u4f7f\u3048\u3070\u5373\u5fae\u5206\u3092\u5b9f\u884c\u3057\u307e\u3059\u3002<\/p>\n<\/div>\n<\/div>\n<\/div>\n
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\u4f8b\uff1a<\/p>\n
$y = x^3 + 5 x + 2$ \u3092 $x$ \u3067\u5fae\u5206\u3057\u307e\u3059\u3002\u3068\u3063\u3068\u3068\u7b54\u3048\u3060\u3051\u3092\u8868\u793a\u3055\u305b\u308b\u306b\u306f…<\/p>\n<\/div>\n<\/div>\n<\/div>\n
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In\u00a0[2]:<\/div>\n
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diff<\/span>(<\/span>x<\/span>**<\/span>3<\/span> +<\/span> 5<\/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[2]:<\/div>\n
$\\displaystyle 3 x^{2} + 5$<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n
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\u4f55\u3092\u8a08\u7b97\u3055\u305b\u308b\u306e\u304b\u3092\u8868\u793a\u3055\u305b\u3066\u304b\u3089\u7b54\u3048\u3092\u8868\u793a\u3055\u305b\u308b\u306b\u306f…<\/p>\n<\/div>\n<\/div>\n<\/div>\n
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In\u00a0[3]:<\/div>\n
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y<\/span> =<\/span> x<\/span>**<\/span>3<\/span> +<\/span> 5<\/span>*<\/span>x<\/span> +<\/span> 2<\/span>\r\n\r\nEq<\/span>(<\/span>Derivative<\/span>(<\/span>y<\/span>,<\/span> x<\/span>),<\/span> \r\n   Derivative<\/span>(<\/span>y<\/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[3]:<\/div>\n
$\\displaystyle \\frac{d}{d x} \\left(x^{3} + 5 x + 2\\right) = 3 x^{2} + 5$<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n
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$y = x^3 + 5 x + 2$ \u3092 $x$ \u30672\u968e\u5fae\u5206\u3057\u307e\u3059\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>Derivative<\/span>(<\/span>y<\/span>,<\/span> x<\/span>,<\/span> 2<\/span>),<\/span> diff<\/span>(<\/span>y<\/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[4]:<\/div>\n
$\\displaystyle \\frac{d^{2}}{d x^{2}} \\left(x^{3} + 5 x + 2\\right) = 6 x$<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n
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\u3079\u304d\u95a2\u6570\uff0c\u5e73\u65b9\u6839\u306e\u5fae\u5206<\/h4>\n
\u3079\u304d\u95a2\u6570 $y = x^p$ \u306e\u5fae\u5206<\/p>\n<\/div>\n<\/div>\n<\/div>\n
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In\u00a0[5]:<\/div>\n
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diff<\/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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Out[5]:<\/div>\n
$\\displaystyle \\frac{p x^{p}}{x}$<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n
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\u76f4\u524d\u306e\u7d50\u679c\u3092 _<\/code> \u3067\u53c2\u7167\u3057\u3066\u7c21\u5358\u5316 simplify()<\/code> \u3057\u3066\u307f\u307e\u3059\u3002<\/p>\n<\/div>\n<\/div>\n<\/div>\n
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In\u00a0[6]:<\/div>\n
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simplify<\/span>(<\/span>_<\/span>)<\/span>\r\n<\/pre>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n
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$\\displaystyle p x^{p – 1}$<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n
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\u3068\u3044\u3046\u3053\u3068\u3067\uff0c<\/p>\n<\/div>\n<\/div>\n<\/div>\n
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In\u00a0[7]:<\/div>\n
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Eq<\/span>(<\/span>Derivative<\/span>(<\/span>x<\/span>**<\/span>p<\/span>,<\/span> x<\/span>),<\/span> diff<\/span>(<\/span>x<\/span>**<\/span>p<\/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[7]:<\/div>\n
$\\displaystyle \\frac{\\partial}{\\partial x} x^{p} = p x^{p – 1}$<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n
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\u504f\u5fae\u5206\u8a18\u53f7\u304c\u51fa\u3066\u304f\u308b\u306e\u306f\uff0c$x$ \u306e\u4ed6\u306b $p$ \u3082\u300c\u5909\u6570\u300d\u3068\u307f\u306a\u3055\u308c\u3066\u3044\u308b\u304b\u3089\u3002<\/p>\n
\u7279\u306b\u5e73\u65b9\u6839 $\\sqrt{x}$ \u306e\u5fae\u5206\u306f…<\/p>\n
$\\displaystyle \\frac{d}{dx} \\sqrt{x} =$ diff(sqrt(x), x)<\/code><\/p>\n<\/div>\n<\/div>\n<\/div>\n
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In\u00a0[8]:<\/div>\n
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Eq<\/span>(<\/span>diff<\/span>(<\/span>sqrt<\/span>(<\/span>x<\/span>),<\/span> x<\/span>,<\/span> evaluate<\/span> =<\/span> False<\/span>),<\/span> \r\n   diff<\/span>(<\/span>sqrt<\/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[8]:<\/div>\n
$\\displaystyle \\frac{d}{d x} \\sqrt{x} = \\frac{1}{2 \\sqrt{x}}$<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n
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\u6307\u6570\u95a2\u6570\u306e\u5fae\u5206<\/h4>\n
\u30cd\u30a4\u30d4\u30a2\u6570 $e$ \u3092\u5e95\u3068\u3057\u305f\u6307\u6570\u95a2\u6570 $e^x =$ exp(x)<\/code> \u306e\u5fae\u5206<\/p>\n
$\\displaystyle \\frac{d}{dx} e^x = $ diff(exp(x), x)<\/code><\/p>\n<\/div>\n<\/div>\n<\/div>\n
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In\u00a0[9]:<\/div>\n
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diff<\/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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$\\displaystyle e^{x}$<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n
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\u5bfe\u6570\u95a2\u6570\u306e\u5fae\u5206<\/h4>\n
\u81ea\u7136\u5bfe\u6570 log(x)<\/code> \u306e\u5fae\u5206<\/p>\n
$\\displaystyle \\frac{d}{dx} \\log x =$ diff(log(x), x)<\/code><\/p>\n<\/div>\n<\/div>\n<\/div>\n
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In\u00a0[10]:<\/div>\n
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diff<\/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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$\\displaystyle \\frac{1}{x}$<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n
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\u4e09\u89d2\u95a2\u6570\u306e\u5fae\u5206<\/h4>\n
$\\sin x, \\ \\cos x, \\ \\tan x$ \u306e\u5fae\u5206<\/p>\n<\/div>\n<\/div>\n<\/div>\n
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In\u00a0[11]:<\/div>\n
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Eq<\/span>(<\/span>Derivative<\/span>(<\/span>sin<\/span>(<\/span>x<\/span>),<\/span> x<\/span>),<\/span> diff<\/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[11]:<\/div>\n
$\\displaystyle \\frac{d}{d x} \\sin{\\left(x \\right)} = \\cos{\\left(x \\right)}$<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n
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In\u00a0[12]:<\/div>\n
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Eq<\/span>(<\/span>Derivative<\/span>(<\/span>cos<\/span>(<\/span>x<\/span>),<\/span> x<\/span>),<\/span> diff<\/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[12]:<\/div>\n
$\\displaystyle \\frac{d}{d x} \\cos{\\left(x \\right)} = – \\sin{\\left(x \\right)}$<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n
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In\u00a0[13]:<\/div>\n
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Eq<\/span>(<\/span>Derivative<\/span>(<\/span>tan<\/span>(<\/span>x<\/span>),<\/span> x<\/span>),<\/span> diff<\/span>(<\/span>tan<\/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[13]:<\/div>\n
$\\displaystyle \\frac{d}{d x} \\tan{\\left(x \\right)} = \\tan^{2}{\\left(x \\right)} + 1$<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n
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\u53f3\u8fba\u3092\u7c21\u5358\u5316\u3057\u307e\u3059\u3002
\ndiff(tan(x), x).simplify()<\/code> \u306f simplify(diff(tan(x), x))<\/code> \u3068\u540c\u3058\u610f\u5473\u3067\u3059\u3002<\/p>\n<\/div>\n<\/div>\n<\/div>\n
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In\u00a0[14]:<\/div>\n
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Eq<\/span>(<\/span>Derivative<\/span>(<\/span>tan<\/span>(<\/span>x<\/span>),<\/span> x<\/span>),<\/span> diff<\/span>(<\/span>tan<\/span>(<\/span>x<\/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[14]:<\/div>\n
$\\displaystyle \\frac{d}{d x} \\tan{\\left(x \\right)} = \\frac{1}{\\cos^{2}{\\left(x \\right)}}$<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n
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\u9006\u4e09\u89d2\u95a2\u6570\u306e\u5fae\u5206<\/h4>\n
$\\displaystyle \\frac{d}{dx} \\sin^{-1} x = \\frac{d}{dx} \\arcsin x = $ diff(asin(x), x)<\/code><\/p>\n<\/div>\n<\/div>\n<\/div>\n
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In\u00a0[15]:<\/div>\n
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Eq<\/span>(<\/span>Derivative<\/span>(<\/span>asin<\/span>(<\/span>x<\/span>),<\/span> x<\/span>),<\/span> diff<\/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[15]:<\/div>\n
$\\displaystyle \\frac{d}{d x} \\operatorname{asin}{\\left(x \\right)} = \\frac{1}{\\sqrt{1 – x^{2}}}$<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n
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\u25cb\u7df4\u7fd2\uff1a$\\arccos x, \\ \\arctan x$ \u306e\u5fae\u5206<\/h4>\n
$\\displaystyle \\frac{d}{dx} \\cos^{-1} x = \\frac{d}{dx} \\arccos x = $ diff(acos(x), x)<\/code> \u3084<\/p>\n
$\\displaystyle \\frac{d}{dx} \\tan^{-1} x = \\frac{d}{dx} \\arctan x = $ diff(atan(x), x)<\/code> \u306b\u3064\u3044\u3066\u3082\u540c\u69d8\u306b\u3084\u3063\u3066\u307f\u3066\u304f\u3060\u3055\u3044\u3002<\/p>\n<\/div>\n<\/div>\n<\/div>\n
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In\u00a0[\u00a0]:<\/div>\n
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In\u00a0[\u00a0]:<\/div>\n
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\u53cc\u66f2\u7dda\u95a2\u6570\u306e\u5fae\u5206<\/h4>\n
$\\displaystyle \\frac{d}{dx} \\cosh x = $ diff(cosh(x), x)<\/code><\/p>\n<\/div>\n<\/div>\n<\/div>\n
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In\u00a0[16]:<\/div>\n
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Eq<\/span>(<\/span>Derivative<\/span>(<\/span>cosh<\/span>(<\/span>x<\/span>),<\/span> x<\/span>),<\/span> diff<\/span>(<\/span>cosh<\/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[16]:<\/div>\n
$\\displaystyle \\frac{d}{d x} \\cosh{\\left(x \\right)} = \\sinh{\\left(x \\right)}$<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n
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\u25cb\u7df4\u7fd2\uff1a$\\sinh x, \\ \\tanh x$ \u306e\u5fae\u5206<\/h4>\n
$\\displaystyle \\frac{d}{dx} \\sinh x = $ diff(sinh(x), x)<\/code> \u3084<\/p>\n
$\\displaystyle \\frac{d}{dx} \\tanh x = $ diff(tanh(x), x)<\/code> \u306b\u3064\u3044\u3066\u3082\u540c\u69d8\u306b\u3002<\/p>\n<\/div>\n<\/div>\n<\/div>\n
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In\u00a0[\u00a0]:<\/div>\n
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In\u00a0[\u00a0]:<\/div>\n
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\u9006\u53cc\u66f2\u7dda\u95a2\u6570\u306e\u5fae\u5206<\/h4>\n
$\\displaystyle \\frac{d}{dx} \\cosh^{-1} x = \\frac{d}{dx} \\mbox{arcosh}\\ x =$ diff(acosh(x), x)<\/code> $\\displaystyle = \\frac{1}{\\sqrt{x^2-1}}$<\/p>\n<\/div>\n<\/div>\n<\/div>\n
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In\u00a0[17]:<\/div>\n
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Eq<\/span>(<\/span>Derivative<\/span>(<\/span>acosh<\/span>(<\/span>x<\/span>),<\/span> x<\/span>),<\/span> 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[17]:<\/div>\n
$\\displaystyle \\frac{d}{d x} \\operatorname{acosh}{\\left(x \\right)} = \\frac{1}{\\sqrt{x – 1} \\sqrt{x + 1}}$<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n
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\u672c\u7a3f\u57f7\u7b46\u6642\u70b9\u3067\u306f\uff0cSymPy \u306f $\\displaystyle \\frac{1}{\\sqrt{x^2-1}}$ \u3067\u306f\u306a\u304f\uff0c\u3042\u3048\u3066 $\\displaystyle \\frac{1}{\\sqrt{x-1}\\sqrt{x+1}}$ \u3068\u8fd4\u3059\u3088\u3046\u3067\u3059\u3002\uff08\u79c1\u898b\u3067\u3059\u304c\uff0c\u3053\u308c\u304c\u5f8c\u3005\uff0c\u3044\u304f\u3064\u304b\u306e\u6b8b\u5ff5\u306a\u7d50\u679c\u3092\u5f15\u304d\u8d77\u3053\u3057\u307e\u3059\u3002\uff09<\/p>\n<\/div>\n<\/div>\n<\/div>\n
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\u25cb\u7df4\u7fd2\uff1a$\\sinh^{-1} x, \\ \\tanh^{-1} x$ \u306e\u5fae\u5206<\/h4>\n
$\\displaystyle \\frac{d}{dx} \\sinh^{-1} x$ \u3084 $\\displaystyle \\frac{d}{dx} \\tanh^{-1} x$ \u306b\u3064\u3044\u3066\u3082\u540c\u69d8\u306b\u5fae\u5206\u3057\u3066\u307f\u3066\u3002<\/p>\n<\/div>\n<\/div>\n<\/div>\n
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In\u00a0[\u00a0]:<\/div>\n
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In\u00a0[\u00a0]:<\/div>\n
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1\u5909\u6570\u95a2\u6570\u306e\u7a4d\u5206 integrate()<\/code><\/h3>\n
\u4e0d\u5b9a\u7a4d\u5206<\/h4>\n
SymPy \u30671\u5909\u6570\u95a2\u6570 $y = f(x)$ \u306e\u7a4d\u5206\u306b\u306f integrate()<\/code> \u3092\u4f7f\u3044\u307e\u3059\u3002<\/p>\n
$\\displaystyle\\int f(x)\\, dx = $ integrate(f(x), x)<\/code> \uff08$=$ Integral(f(x), x).doit()<\/code>\uff09<\/p>\n
\u4e0d\u5b9a\u7a4d\u5206\u306e\u969b\uff0c\u7a4d\u5206\u5b9a\u6570\u304c\u7701\u7565\u3055\u308c\u305f\u7b54\u3048\u304c\u51fa\u529b\u3055\u308c\u307e\u3059\u3002<\/p>\n
\u5b9a\u7a4d\u5206<\/h4>\n
\u5b9a\u7a4d\u5206\u306e\u969b\u306f\u4ee5\u4e0b\u306e\u3088\u3046\u306b\u7a4d\u5206\u7bc4\u56f2\u3092 (x, a, b)<\/code> \u306e\u3088\u3046\u306b\u6307\u5b9a\u3057\u307e\u3059\u3002<\/p>\n
$\\displaystyle\\int_a^b f(x)\\, dx = $ integrate(f(x), (x, a, b))<\/code><\/p>\n<\/div>\n<\/div>\n<\/div>\n
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\u53c2\u8003\uff1aIntegral()<\/code> \u3068 integrate()<\/code><\/h4>\n
Integral()<\/code> \u306f\u7a4d\u5206\u306e\u8868\u793a\u3092\u3059\u308b\u3060\u3051\u3067\u5b9f\u969b\u306b\u7a4d\u5206\u306f\u884c\u3044\u307e\u305b\u3093\u3002\u7a4d\u5206\u3092\u5b9f\u884c\u3059\u308b\u306b\u306f\uff0cIntegral().doit()<\/code> \u306e\u3088\u3046\u306b .doit()<\/code> \u3092\u8ffd\u52a0\u3057\u307e\u3059\u3002<\/p>\n
integrate()<\/code> \u3092\u4f7f\u3048\u3070\u5373\u7a4d\u5206\u3092\u5b9f\u884c\u3057\u307e\u3059\u3002<\/p>\n<\/div>\n<\/div>\n<\/div>\n
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\u3079\u304d\u95a2\u6570\u306e\u4e0d\u5b9a\u7a4d\u5206<\/h4>\n
$\\displaystyle \\int x^p\\, dx =$ integrate(x**p, x)<\/code><\/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>x<\/span>**<\/span>p<\/span>,<\/span> x<\/span>),<\/span> integrate<\/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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Out[18]:<\/div>\n
$\\displaystyle \\int x^{p}\\, dx = \\begin{cases} \\frac{x^{p + 1}}{p + 1} & \\text{for}\\: p \\neq -1 \\\\\\log{\\left(x \\right)} & \\text{otherwise} \\end{cases}$<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n
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$p=-1$ \u306e\u3068\u304d\u306f\uff0c\u4ee5\u4e0b\u306e\u300c\u5bfe\u6570\u95a2\u6570\u306b\u95a2\u9023\u3057\u305f\u4e0d\u5b9a\u7a4d\u5206\u300d\u3082\u53c2\u7167\uff1a<\/p>\n<\/div>\n<\/div>\n<\/div>\n
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\u6307\u6570\u95a2\u6570\u306e\u4e0d\u5b9a\u7a4d\u5206<\/h4>\n
$\\displaystyle \\int e^x\\, dx = $ integrate(exp(x), x)<\/code><\/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>exp<\/span>(<\/span>x<\/span>),<\/span> x<\/span>),<\/span> 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[19]:<\/div>\n
$\\displaystyle \\int e^{x}\\, dx = e^{x}$<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n
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\u5bfe\u6570\u95a2\u6570\u306b\u95a2\u9023\u3057\u305f\u4e0d\u5b9a\u7a4d\u5206<\/h4>\n
$\\displaystyle \\int \\frac{1}{x}\\, dx = \\log |x| + C$ \u3068\u306a\u308b\u306f\u305a\u3067\u3059\u304c…<\/p>\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>1<\/span>\/<\/span>x<\/span>,<\/span> x<\/span>),<\/span> integrate<\/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[20]:<\/div>\n
$\\displaystyle \\int \\frac{1}{x}\\, dx = \\log{\\left(x \\right)}$<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n
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SymPy \u306e\u7a4d\u5206 integrate()<\/code> \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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\u25cb\u7df4\u7fd2\uff1a$\\log x$ \u306e\u4e0d\u5b9a\u7a4d\u5206<\/h4>\n
$\\displaystyle \\int \\log x\\, dx = $ integrate(log(x), x)<\/code> $= \\cdots$<\/p>\n<\/div>\n<\/div>\n<\/div>\n
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In\u00a0[\u00a0]:<\/div>\n
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\u4e0a\u8a18\u306e\u7d50\u679c\u306f\uff0c\u3044\u304b\u306b\u3082\u90e8\u5206\u7a4d\u5206\u3092\u3057\u3066\u7b54\u3048\u3092\u51fa\u3057\u3066\u3044\u308b\u3088\u3046\u3067\u3059\u306d\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[21]:<\/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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Out[21]:<\/div>\n
$\\displaystyle \\int \\cos{\\left(x \\right)}\\, dx = \\sin{\\left(x \\right)}$<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n
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In\u00a0[22]:<\/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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Out[22]:<\/div>\n
$\\displaystyle \\int \\sin{\\left(x \\right)}\\, dx = – \\cos{\\left(x \\right)}$<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n
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In\u00a0[23]:<\/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   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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Out[23]:<\/div>\n
$\\displaystyle \\int \\frac{1}{\\cos^{2}{\\left(x \\right)}}\\, dx = \\tan{\\left(x \\right)}$<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n
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\u25cb\u7df4\u7fd2\uff1a$\\tan x$ \u306e\u4e0d\u5b9a\u7a4d\u5206<\/h4>\n
\u3067\u306f\uff0c$\\displaystyle \\int \\tan x\\, dx $ \u306f\uff1f<\/p>\n<\/div>\n<\/div>\n<\/div>\n
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In\u00a0[\u00a0]:<\/div>\n
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\u9006\u4e09\u89d2\u95a2\u6570\u306b\u95a2\u9023\u3057\u305f\u4e0d\u5b9a\u7a4d\u5206<\/h4>\n<\/div>\n<\/div>\n<\/div>\n
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$\\displaystyle \\frac{d}{dx} \\arcsin x = \\frac{1}{\\sqrt{1-x^2}}$ \u3060\u3063\u305f\u306e\u3067\uff0c<\/p>\n
$\\displaystyle \\int \\frac{1}{\\sqrt{1-x^2}}\\, dx = \\arcsin x + C$ \u3068\u306a\u308b\u306f\u305a\u3002<\/p>\n<\/div>\n<\/div>\n<\/div>\n
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In\u00a0[24]:<\/div>\n
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Eq<\/span>(<\/span>Integral<\/span>(<\/span>1<\/span>\/<\/span>(<\/span>sqrt<\/span>(<\/span>1<\/span> -<\/span> x<\/span>**<\/span>2<\/span>)),<\/span> x<\/span>),<\/span> \r\n   integrate<\/span>(<\/span>1<\/span>\/<\/span>(<\/span>sqrt<\/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[24]:<\/div>\n
$\\displaystyle \\int \\frac{1}{\\sqrt{1 – x^{2}}}\\, dx = \\operatorname{asin}{\\left(x \\right)}$<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n
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$\\displaystyle \\frac{d}{dx} \\arccos x = -\\frac{1}{\\sqrt{1-x^2}}$ \u3060\u3063\u305f\u306e\u3067\uff0c<\/p>\n
$\\displaystyle \\int \\frac{-1}{\\sqrt{1-x^2}}\\, dx = \\arccos x + C$ \u3068\u306a\u3063\u3066\u3082\u3044\u3044\u306f\u305a\u3060\u304c…<\/p>\n<\/div>\n<\/div>\n<\/div>\n
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In\u00a0[25]:<\/div>\n
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Eq<\/span>(<\/span>Integral<\/span>(<\/span>-<\/span>1<\/span>\/<\/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>(<\/span>sqrt<\/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[25]:<\/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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\u7a4d\u5206\u306e\u7b54\u3048\u304c $-\\arcsin x$ \u3068\u51fa\u3066\u3082\uff0c\u614c\u3066\u305a\u9a12\u304c\u305a\uff0c\u76f4\u8fd1\u306e\u51fa\u529b _<\/code> \u3092 .rewrite(acos)<\/code> \u3064\u307e\u308a $\\arccos x$ \u3092\u4f7f\u3063\u3066\u66f8\u304d\u76f4\u3057\u3066\u307f\u308b\u3068…<\/p>\n<\/div>\n<\/div>\n<\/div>\n
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In\u00a0[26]:<\/div>\n
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_<\/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[26]:<\/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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$\\displaystyle -\\frac{\\pi}{2}$ \u306f\u7a4d\u5206\u5b9a\u6570 $C$ \u306b\u7d44\u307f\u8fbc\u307e\u308c\u308b\u5b9a\u6570\u306a\u306e\u3067\uff0c\u7d50\u679c\uff0c<\/p>\n
$\\displaystyle \\int \\frac{-1}{\\sqrt{1-x^2}}\\, dx = \\arccos x + C$ \u3068\u306a\u3063\u3066\u3082\u3044\u3044\u3053\u3068\u306b\u306a\u308b\u3002<\/p>\n<\/div>\n<\/div>\n<\/div>\n
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$\\displaystyle \\frac{d}{dx} \\arctan x = \\frac{1}{1+x^2}$ \u3060\u3063\u305f\u306e\u3067\uff0c<\/p>\n
$\\displaystyle \\int \\frac{1}{1+x^2}\\, dx = \\arctan x + C$ \u3068\u306a\u308b\u306f\u305a\u3002<\/p>\n<\/div>\n<\/div>\n<\/div>\n
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In\u00a0[27]:<\/div>\n
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integrate<\/span>(<\/span>1<\/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[27]:<\/div>\n
$\\displaystyle \\operatorname{atan}{\\left(x \\right)}$<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n
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\u25cb\u7df4\u7fd2\uff1a\u9006\u4e09\u89d2\u95a2\u6570\u306e\u4e0d\u5b9a\u7a4d\u5206<\/h4>\n
$\\displaystyle \\int \\arcsin x\\, dx, \\ \\int \\arccos x\\, dx, \\ \\int \\arctan x\\, dx$ \u306b\u3064\u3044\u3066\u3082\u8a08\u7b97\u3057\u3066\u307f\u3088\u3002<\/p>\n<\/div>\n<\/div>\n<\/div>\n
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In\u00a0[\u00a0]:<\/div>\n
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In\u00a0[\u00a0]:<\/div>\n
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In\u00a0[\u00a0]:<\/div>\n
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\u53cc\u66f2\u7dda\u95a2\u6570\u306e\u4e0d\u5b9a\u7a4d\u5206<\/h4>\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>cosh<\/span>(<\/span>x<\/span>),<\/span> x<\/span>),<\/span> integrate<\/span>(<\/span>cosh<\/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
$\\displaystyle \\int \\cosh{\\left(x \\right)}\\, dx = \\sinh{\\left(x \\right)}$<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n
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In\u00a0[29]:<\/div>\n
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Eq<\/span>(<\/span>Integral<\/span>(<\/span>sinh<\/span>(<\/span>x<\/span>),<\/span> x<\/span>),<\/span> integrate<\/span>(<\/span>sinh<\/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
$\\displaystyle \\int \\sinh{\\left(x \\right)}\\, dx = \\cosh{\\left(x \\right)}$<\/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>1<\/span>\/<\/span>cosh<\/span>(<\/span>x<\/span>)<\/span>**<\/span>2<\/span>,<\/span> x<\/span>),<\/span> integrate<\/span>(<\/span>1<\/span>\/<\/span>cosh<\/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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Out[30]:<\/div>\n
$\\displaystyle \\int \\frac{1}{\\cosh^{2}{\\left(x \\right)}}\\, dx = \\tanh{\\left(x \\right)}$<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n
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\u25cb\u7df4\u7fd2\uff1a$\\tanh x $ \u306e\u4e0d\u5b9a\u7a4d\u5206<\/h4>\n
\u3067\u306f $\\displaystyle \\int \\tanh x\\, dx$ \u306f\uff1f<\/p>\n<\/div>\n<\/div>\n<\/div>\n
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In\u00a0[\u00a0]:<\/div>\n
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\u9006\u53cc\u66f2\u7dda\u95a2\u6570\u306b\u95a2\u9023\u3057\u305f\u4e0d\u5b9a\u7a4d\u5206<\/h4>\n<\/div>\n<\/div>\n<\/div>\n
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$\\displaystyle 1. \\ \\int \\frac{1}{\\sqrt{x^2-1}}\\, dx$<\/p>\n
$\\displaystyle \\frac{d}{dx} \\cosh^{-1} x = \\frac{d}{dx} \\mbox{arcosh}\\ x =$ diff(acosh(x), x)<\/code> $\\displaystyle = \\frac{1}{\\sqrt{x^2-1}}$ \u306a\u306e\u3067<\/p>\n
$\\displaystyle \\int \\frac{1}{\\sqrt{x^2-1}}\\, dx = \\cosh^{-1} x$ \u3068\u306a\u308b\u306f\u305a\u3060\u304c…<\/p>\n<\/div>\n<\/div>\n<\/div>\n
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In\u00a0[31]:<\/div>\n
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Eq<\/span>(<\/span>Integral<\/span>(<\/span>1<\/span>\/<\/span>sqrt<\/span>(<\/span>x<\/span>**<\/span>2<\/span> -<\/span> 1<\/span>),<\/span> x<\/span>),<\/span> \r\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
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Out[31]:<\/div>\n
$\\displaystyle \\int \\frac{1}{\\sqrt{x^{2} – 1}}\\, dx = \\log{\\left(x + \\sqrt{x^{2} – 1} \\right)}$<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n
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$\\displaystyle \\cosh^{-1} x \\Rightarrow \\log\\left(x + \\sqrt{x^2-1}\\right)$ \u3068\u66f8\u304d\u76f4\u3057\u3066\u306f\u304f\u308c\u308b\u304c…<\/p>\n<\/div>\n<\/div>\n<\/div>\n
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In\u00a0[32]:<\/div>\n
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acosh<\/span>(<\/span>x<\/span>)<\/span>.<\/span>rewrite<\/span>(<\/span>log<\/span>)<\/span>.<\/span>factor<\/span>()<\/span>\r\n<\/pre>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n
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Out[32]:<\/div>\n
$\\displaystyle \\log{\\left(x + \\sqrt{x – 1} \\sqrt{x + 1} \\right)}$<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n
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$\\displaystyle \\log\\left(x + \\sqrt{x^2-1}\\right) \\Rightarrow \\cosh^{-1} x $ \u3068\u306f\u66f8\u304d\u76f4\u3057\u3066\u304f\u308c\u306a\u3044\u3002<\/p>\n<\/div>\n<\/div>\n<\/div>\n
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In\u00a0[33]:<\/div>\n
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log<\/span>(<\/span>x<\/span> +<\/span> sqrt<\/span>(<\/span>x<\/span>**<\/span>2<\/span> -<\/span> 1<\/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[33]:<\/div>\n
$\\displaystyle \\log{\\left(x + \\sqrt{x^{2} – 1} \\right)}$<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n
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$\\displaystyle 2. \\ \\int \\frac{1}{\\sqrt{1+x^2}}\\, dx$<\/p>\n
$\\displaystyle \\frac{d}{dx} \\sinh^{-1} x = \\frac{1}{\\sqrt{1+x^2}}$ \u306a\u306e\u3067…<\/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>1<\/span>\/<\/span>sqrt<\/span>(<\/span>1<\/span> +<\/span> x<\/span>**<\/span>2<\/span>),<\/span> x<\/span>),<\/span> \r\n   integrate<\/span>(<\/span>1<\/span>\/<\/span>sqrt<\/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[34]:<\/div>\n
$\\displaystyle \\int \\frac{1}{\\sqrt{x^{2} + 1}}\\, dx = \\operatorname{asinh}{\\left(x \\right)}$<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n
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$\\displaystyle 3. \\ \\int \\frac{1}{1-x^2}\\, dx$<\/p>\n
$\\displaystyle \\frac{d}{dx} \\tanh^{-1} x = \\frac{1}{1-x^2}$ \u306a\u306e\u3067<\/p>\n
$\\displaystyle \\int \\frac{1}{1-x^2}\\, dx = \\tanh^{-1} x + C$ \u3068\u306a\u308b\u306f\u305a\u3060\u304c…<\/p>\n<\/div>\n<\/div>\n<\/div>\n
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In\u00a0[35]:<\/div>\n
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Eq<\/span>(<\/span>Integral<\/span>(<\/span>1<\/span>\/<\/span>(<\/span>1<\/span>-<\/span>x<\/span>**<\/span>2<\/span>),<\/span> x<\/span>),<\/span> integrate<\/span>(<\/span>1<\/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[35]:<\/div>\n
$\\displaystyle \\int \\frac{1}{1 – x^{2}}\\, dx = – \\frac{\\log{\\left(x – 1 \\right)}}{2} + \\frac{\\log{\\left(x + 1 \\right)}}{2}$<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n
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$\\displaystyle 4. \\ \\int \\sinh^{-1} x\\, dx$<\/p>\n<\/div>\n<\/div>\n<\/div>\n
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In\u00a0[36]:<\/div>\n
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Eq<\/span>(<\/span>Integral<\/span>(<\/span>asinh<\/span>(<\/span>x<\/span>),<\/span> x<\/span>),<\/span> 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[36]:<\/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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$\\displaystyle 5. \\ \\int \\tanh^{-1} x\\, dx$<\/p>\n<\/div>\n<\/div>\n<\/div>\n
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In\u00a0[37]:<\/div>\n
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Eq<\/span>(<\/span>Integral<\/span>(<\/span>atanh<\/span>(<\/span>x<\/span>),<\/span> x<\/span>),<\/span> 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[37]:<\/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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$\\displaystyle 6. \\ \\int \\cosh^{-1} x\\, dx$<\/p>\n<\/div>\n<\/div>\n<\/div>\n
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In\u00a0[38]:<\/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[38]:<\/div>\n
$\\displaystyle \\int \\operatorname{acosh}{\\left(x \\right)}\\, dx$<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n
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\u4e0a\u8a18\u306e\u3088\u3046\u306b\uff0c\u672c\u7a3f\u57f7\u7b46\u6642\u70b9\u3067\u306f\uff0cSymPy \u306f $\\cosh^{-1} x$ \u306e\u7a4d\u5206\u304c\u3067\u304d\u306a\u3044\u3002\u3053\u308c\u306f\u90e8\u5206\u7a4d\u5206\u3067\u7c21\u5358\u306b\u6c42\u307e\u308b\u304b\u3089\uff0c\u4ee5\u4e0b\u306e\u3088\u3046\u306b\u4eba\u529b\u3067\u8a08\u7b97\u3057\u3066\u307f\u3088\u3046\u3002<\/p>\n
\\begin{eqnarray}
\n\\int \\cosh^{-1} x\\, dx &=& \\int (x^{\\prime})\\cdot\\cosh^{-1} x\\, dx \\\\
\n&=& x \\, \\cosh^{-1} x – \\int x\\, \\left( \\cosh^{-1} x \\right)^{\\prime}\\, dx \\\\
\n% &=& x \\, \\cosh^{-1} x – \\int x\\, \\frac{1}{\\sqrt{x^2-1}}\\, dx \\\\
\n&=& \\cdots
\n\\end{eqnarray}<\/p>\n
SymPy \u304c\u3053\u306e\u7a4d\u5206\u304c\u3067\u304d\u306a\u3044\u306e\u306f\uff0c\u3069\u3046\u3082\u4ee5\u4e0b\u306e\u3088\u3046\u306b…<\/p>\n<\/div>\n<\/div>\n<\/div>\n
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In\u00a0[39]:<\/div>\n
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Eq<\/span>(<\/span>Derivative<\/span>(<\/span>acosh<\/span>(<\/span>x<\/span>),<\/span> x<\/span>),<\/span> 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[39]:<\/div>\n
$\\displaystyle \\frac{d}{d x} \\operatorname{acosh}{\\left(x \\right)} = \\frac{1}{\\sqrt{x – 1} \\sqrt{x + 1}}$<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n
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$\\displaystyle \\frac{d}{dx} \\cosh^{-1} x$ \u3092 $\\displaystyle\\frac{1}{\\sqrt{x^2-1}}$ \u3067\u306f\u306a\u304f $\\displaystyle\\frac{1}{\\sqrt{x-1}\\sqrt{x+1}}$ \u3068\u3059\u308b\u3053\u3068\u306b\u3042\u308b\u306e\u3067\u306f\uff1f\u3068\u63a8\u6e2c\u3059\u308b\u3002<\/p>\n
\u305f\u3081\u3057\u306b\uff0c$\\displaystyle \\int \\frac{x}{\\sqrt{x-1}\\sqrt{x+1}}\\, dx$ \u3092\u8a08\u7b97\u3057\u3066\u307f\u308b\u3068…<\/p>\n<\/div>\n<\/div>\n<\/div>\n
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In\u00a0[40]:<\/div>\n
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integrate<\/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[40]:<\/div>\n
$\\displaystyle \\frac{{G_{6, 6}^{6, 2}\\left(\\begin{matrix} – \\frac{1}{4}, \\frac{1}{4} & 0, 0, \\frac{1}{2}, 1 \\\\- \\frac{1}{2}, – \\frac{1}{4}, 0, \\frac{1}{4}, \\frac{1}{2}, 0 & \\end{matrix} \\middle| {\\frac{1}{x^{2}}} \\right)}}{4 \\pi^{\\frac{3}{2}}} + \\frac{i {G_{6, 6}^{2, 6}\\left(\\begin{matrix} -1, – \\frac{3}{4}, – \\frac{1}{2}, – \\frac{1}{4}, 0, 1 & \\\\- \\frac{3}{4}, – \\frac{1}{4} & -1, – \\frac{1}{2}, – \\frac{1}{2}, 0 \\end{matrix} \\middle| {\\frac{e^{2 i \\pi}}{x^{2}}} \\right)}}{4 \\pi^{\\frac{3}{2}}}$<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n
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\u306a\u3093\u304b\u610f\u5473\u4e0d\u660e\u306e\u7b54\u3048\u304c\u51fa\u3066\u304f\u308b\u3067\u3042\u308d\u3046\u3002<\/p>\n
\u4e00\u65b9\uff0c$\\displaystyle \\int \\frac{x}{\\sqrt{x^2-1}}\\, dx$ \u3092\u8a08\u7b97\u3057\u3066\u307f\u308b\u3068…<\/p>\n<\/div>\n<\/div>\n<\/div>\n
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In\u00a0[41]:<\/div>\n
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Eq<\/span>(<\/span>Integral<\/span>(<\/span>x<\/span>\/<\/span>sqrt<\/span>(<\/span>x<\/span>**<\/span>2<\/span>-<\/span>1<\/span>),<\/span> x<\/span>),<\/span> \r\n   integrate<\/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
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Out[41]:<\/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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\u4ee5\u4e0a\u306e\u3053\u3068\u304b\u3089<\/p>\n
\\begin{eqnarray}
\n\\int \\cosh^{-1} x\\, dx
\n&=& x \\, \\cosh^{-1} x – \\int x\\, \\left( \\cosh^{-1} x \\right)^{\\prime}\\, dx \\\\
\n&=& x \\, \\cosh^{-1} x – \\int \\frac{x}{\\sqrt{x^2-1}}\\, dx \\\\
\n&=& \\cdots
\n\\end{eqnarray}<\/p>\n<\/div>\n<\/div>\n<\/div>\n
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\u25cb\u7df4\u7fd2\uff1a$\\cosh^{-1} x$ \u306e\u4e0d\u5b9a\u7a4d\u5206<\/h4>\n
\u90e8\u5206\u7a4d\u5206\u3092\u4f7f\u3063\u3066\uff0c\u3042\u3089\u305f\u3081\u3066 $\\displaystyle \\int \\cosh^{-1} x\\, dx $ \u3092\u6c42\u3081\u3088\u3002<\/p>\n<\/div>\n<\/div>\n<\/div>\n
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In\u00a0[\u00a0]:<\/div>\n
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\u504f\u5fae\u5206\uff1a\u591a\u5909\u6570\uff08\u4e3b\u306b2\u5909\u6570\uff09\u95a2\u6570\u306e\u5fae\u5206<\/h3>\n
\u4e3b\u306b2\u5909\u6570\u95a2\u6570 $f(x, y)$ \u306e\u504f\u5fae\u5206\u306b\u3064\u3044\u3066\u307e\u3068\u3081\u3066\u304a\u304d\u307e\u3059\u3002<\/p>\n
1\u968e\u504f\u5fae\u5206 diff()<\/code><\/h4>\n
SymPy \u3067\u306e\u504f\u5fae\u5206\u306f\uff081\u5909\u6570\u95a2\u6570\u306e\u5fae\u5206\u3067\u3042\u308b\u300c\u5e38\u5fae\u5206\u300d\u3068\u540c\u69d8\u306e\u66f8\u304d\u65b9\u3067\uff09\u4ee5\u4e0b\u306e\u3088\u3046\u306b\u66f8\u304d\u307e\u3059\u3002<\/p>\n
$\\displaystyle \\frac{\\partial}{\\partial x} f(x, y) = $ diff(f(x, y), x)<\/code><\/p>\n
$\\displaystyle \\frac{\\partial}{\\partial y} f(x, y) = $ diff(f(x, y), y)<\/code><\/p>\n<\/div>\n<\/div>\n<\/div>\n
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In\u00a0[42]:<\/div>\n
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# \u5909\u6570 x, y \u3092\u521d\u671f\u5316<\/span>\r\nvar<\/span>(<\/span>'x y'<\/span>)<\/span>\r\n\r\n# f \u3092\u672a\u5b9a\u7fa9\u95a2\u6570\u3068\u3057\u3066\u5ba3\u8a00<\/span>\r\nf<\/span> =<\/span> Function<\/span>(<\/span>'f'<\/span>)<\/span>\r\n<\/pre>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n
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In\u00a0[43]:<\/div>\n
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diff<\/span>(<\/span>f<\/span>(<\/span>x<\/span>,<\/span> y<\/span>),<\/span> x<\/span>)<\/span>\r\n<\/pre>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n
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Out[43]:<\/div>\n
$\\displaystyle \\frac{\\partial}{\\partial x} f{\\left(x,y \\right)}$<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n
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In\u00a0[44]:<\/div>\n
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diff<\/span>(<\/span>f<\/span>(<\/span>x<\/span>,<\/span> y<\/span>),<\/span> y<\/span>)<\/span>\r\n<\/pre>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n
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Out[44]:<\/div>\n
$\\displaystyle \\frac{\\partial}{\\partial y} f{\\left(x,y \\right)}$<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n
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\u25cb\u7df4\u7fd2\uff1a\u504f\u5c0e\u95a2\u6570\u3092\u6c42\u3081\u308b<\/h4>\n
\u95a2\u6570 $z = x^3 – 4 x^2 y + x y + 3 y^2$ \u306e\u504f\u5c0e\u95a2\u6570\u3092\u6c42\u3081\u3088\u3002<\/p>\n
\u300c\u504f\u5c0e\u95a2\u6570\u3092\u6c42\u3081\u3088\u300d\u3068\u306f\uff0c$x, y$ \u306e2\u5909\u6570\u95a2\u6570\u3067\u3042\u308b $z$ \u306b\u3064\u3044\u3066 $\\displaystyle \\frac{\\partial z}{\\partial x}$ \u3068
\n$\\displaystyle \\frac{\\partial z}{\\partial y}$ \u3092\u8a08\u7b97\u3057\u3066\u6c42\u3081\u3088\uff0c\u3068\u3044\u3046\u3053\u3068\u3002<\/p>\n<\/div>\n<\/div>\n<\/div>\n
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In\u00a0[\u00a0]:<\/div>\n
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In\u00a0[\u00a0]:<\/div>\n
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\u9ad8\u968e\u504f\u5fae\u5206 diff(f(x, y), x, m, y, n)<\/code><\/h4>\n
2\u968e\u4ee5\u4e0a\u306e\u9ad8\u968e\u504f\u5fae\u5206\u306f\u4ee5\u4e0b\u306e\u3088\u3046\u306b\u66f8\u304d\u307e\u3059\u3002<\/p>\n
$\\displaystyle \\frac{\\partial^2}{\\partial x^2} f(x, y) = $ diff(f(x, y), x, 2)<\/code><\/p>\n
$\\displaystyle \\frac{\\partial}{\\partial y}\\left(\\frac{\\partial}{\\partial x} f(x, y)\\right) = \\frac{\\partial^2}{\\partial y \\partial x} f(x, y) = $ diff(f(x, y), x, y)<\/code><\/p>\n
$\\displaystyle \\frac{\\partial^n}{\\partial y^n}\\left(\\frac{\\partial^m}{\\partial x^m} f(x, y)\\right) = $ diff(f(x, y), x, m, y, n)<\/code><\/p>\n<\/div>\n<\/div>\n<\/div>\n
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\u25cb\u7df4\u7fd2\uff1a\u9ad8\u968e\u5c0e\u95a2\u6570\u306e\u8a08\u7b97<\/h4>\n
$1.\\ $ 2\u5909\u6570\u95a2\u6570 $\\displaystyle z = \\tan^{-1} \\frac{y}{x} = \\arctan \\frac{y}{x}$ \u306b\u3064\u3044\u3066\uff0c\u4ee5\u4e0b\u306e\u9ad8\u968e\u504f\u5c0e\u95a2\u6570\u3092\u8a08\u7b97\u3057\uff0c\u7c21\u5358\u5316\u3057\u305f\u8868\u793a\u3067\u8868\u305b\u3002<\/p>\n
$$(1). \\ \\frac{\\partial^2 z}{\\partial y\\partial x}, \\quad(2). \\ \\frac{\\partial^2 z}{\\partial x\\partial y}, \\quad
\n(3). \\ \\frac{\\partial^2 z}{\\partial x^2}, \\quad(4). \\ \\frac{\\partial^2 z}{\\partial y^2}$$<\/p>\n<\/div>\n<\/div>\n<\/div>\n
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In\u00a0[\u00a0]:<\/div>\n
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In\u00a0[\u00a0]:<\/div>\n
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In\u00a0[\u00a0]:<\/div>\n
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$2. \\ z = \\log\\sqrt{x^2 + y^2}$ \u306e\u3068\u304d\uff0c\u6b21\u306e\u5024\u3092\u6c42\u3081\u3088\u3002<\/p>\n
$$\\frac{\\partial^2 z}{\\partial x^2} + \\frac{\\partial^2 z}{\\partial y^2} = ?$$<\/p>\n<\/div>\n<\/div>\n<\/div>\n
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In\u00a0[\u00a0]:<\/div>\n
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SymPy \u304c2\u968e\u504f\u5c0e\u95a2\u6570\u306b\u95a2\u3057\u3066<\/p>\n
$$\\frac{\\partial^2 f}{\\partial x \\partial y} = \\frac{\\partial^2 f}{\\partial y \\partial x}$$<\/p>\n
\u306e\u3088\u3046\u306b\uff0c\u504f\u5fae\u5206\u306e\u9806\u5e8f\u3092\u5909\u3048\u3066\u3082\u3088\u3044\u3053\u3068\u3092\u77e5\u3063\u3066\u3044\u308b\u304b\u78ba\u8a8d\u3057\u307e\u3059\u3002<\/p>\n
\u4ee5\u4e0b\u3067\u306f\uff0c\u6bd4\u8f03\u6f14\u7b97\u5b50 ==<\/code> \u3092\u4f7f\u3063\u3066\u5de6\u8fba\u3068\u53f3\u8fba\u304c\u7b49\u3057\u3044\u304b\u3092\u78ba\u8a8d\u3057\u3066\u3044\u307e\u3059\u3002\u7b49\u3057\u3051\u308c\u3070 True<\/code>\uff0c\u7b49\u3057\u304f\u306a\u3051\u308c\u3070 False<\/code> \u3092\u8fd4\u3057\u307e\u3059\u3002<\/p>\n<\/div>\n<\/div>\n<\/div>\n
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In\u00a0[45]:<\/div>\n
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diff<\/span>(<\/span>f<\/span>(<\/span>x<\/span>,<\/span> y<\/span>),<\/span> x<\/span>,<\/span> y<\/span>)<\/span> ==<\/span> diff<\/span>(<\/span>f<\/span>(<\/span>x<\/span>,<\/span> y<\/span>),<\/span> y<\/span>,<\/span> x<\/span>)<\/span>\r\n<\/pre>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n
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Out[45]:<\/div>\n
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True<\/pre>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n
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\u4ee5\u4e0b\u3067\u306f\uff0c\u5de6\u8fba\u304b\u3089\u53f3\u8fba\u3092\u5f15\u3044\u3066\u30bc\u30ed\u306b\u306a\u308b\u304b\u3069\u3046\u304b\u3067\uff0c\u78ba\u8a8d\u3057\u3066\u3044\u307e\u3059\u3002<\/p>\n<\/div>\n<\/div>\n<\/div>\n
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In\u00a0[46]:<\/div>\n
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Eq<\/span>(<\/span>Derivative<\/span>(<\/span>f<\/span>(<\/span>x<\/span>,<\/span> y<\/span>),<\/span> x<\/span>,<\/span> y<\/span>)<\/span>-<\/span>Derivative<\/span>(<\/span>f<\/span>(<\/span>x<\/span>,<\/span> y<\/span>),<\/span> y<\/span>,<\/span> x<\/span>),<\/span>\r\n   diff<\/span>(<\/span>f<\/span>(<\/span>x<\/span>,<\/span> y<\/span>),<\/span> x<\/span>,<\/span> y<\/span>)<\/span>-<\/span>diff<\/span>(<\/span>f<\/span>(<\/span>x<\/span>,<\/span> y<\/span>),<\/span> y<\/span>,<\/span> x<\/span>))<\/span>\r\n<\/pre>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n
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Out[46]:<\/div>\n
$\\displaystyle \\frac{\\partial^{2}}{\\partial y\\partial x} f{\\left(x,y \\right)} – \\frac{\\partial^{2}}{\\partial x\\partial y} f{\\left(x,y \\right)} = 0$<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n
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2\u91cd\u7a4d\u5206\uff1a2\u5909\u6570\u95a2\u6570\u306e\u7a4d\u5206<\/h3>\n
\u591a\u5909\u6570\u95a2\u6570\u306e\u7a4d\u5206\u3067\u3042\u308b\u591a\u91cd\u7a4d\u5206\u306e\u3046\u3061\uff0c2\u5909\u6570\u95a2\u6570\u306e\u7a4d\u5206\u3067\u3042\u308b2\u91cd\u7a4d\u5206\u306b\u3064\u3044\u3066\u307e\u3068\u3081\u3066\u304a\u304d\u307e\u3059\u3002<\/p>\n
\u7d2f\u6b21\u7a4d\u5206<\/h4>\n
2\u91cd\u7a4d\u5206\u306e\u8a08\u7b97\u306f\uff0c\u57fa\u672c\u7684\u306b\u7d2f\u6b21\u7a4d\u5206\u3067\u3084\u308a\u307e\u3059\u3002<\/p>\n
$\\displaystyle \\iint_D f(x, y)\\, dx dy$ \u306e\u9818\u57df $D$ \u304c\u4f55\u3068\u4f55\u3067\u56f2\u307e\u308c\u3066\u3044\u308b\u304b\u3092\u660e\u3089\u304b\u306b\u3057\u3066\u7a4d\u5206\u3059\u308b\u3002<\/p>\n
\u30b1\u30fc\u30b9 1<\/h5>\n
\u9818\u57df $D$ \u304c $x_1 \\leq x \\leq x_2,\\ y_1 \\leq y \\leq y_2$ \u3067\u3042\u308b\u5834\u5408\uff1a<\/p>\n
$$\\iint_D f(x, y) \\,dx\\,dy = \\int_{x_1}^{x_2} \\left\\{\\int_{y_1}^{y_2} f(x, y)\\,dy \\right\\} dx$$<\/p>\n
\u307e\u305f\u306f<\/p>\n
$$\\iint_D f(x, y) \\,dx\\,dy = \\int_{y_1}^{y_2} \\left\\{\\int_{x_1}^{x_2} f(x, y)\\,dx \\right\\} dy$$<\/p>\n
\u30b1\u30fc\u30b9 2<\/h5>\n
\u9818\u57df $D$ \u304c $x_1 \\leq x \\leq x_2,\\ y_1(x) \\leq y \\leq y_2(x)$ \u3067\u3042\u308b\u5834\u5408\uff1a<\/p>\n
$$ \\iint_D f(x, y) \\,dx\\,dy = \\int_{x_1}^{x_2} \\left\\{\\int_{y_1(x)}^{y_2(x)} f(x, y)\\,dy \\right\\} dx$$<\/p>\n
\u30b1\u30fc\u30b9 3<\/h5>\n
\u9818\u57df $D$ \u304c $x_1(y)\\leq x \\leq x_2(y),\\ y_1 \\leq y \\leq y_2$ \u3067\u3042\u308b\u5834\u5408\uff1a<\/p>\n
$$ \\iint_D f(x, y) \\,dx\\,dy = \\int_{y_1}^{y_2} \\left\\{\\int_{x_1(y)}^{x_2(y)} f(x, y)\\,dx \\right\\} dy$$<\/p>\n
SymPy \u3067\u306f2\u91cd\u7a4d\u5206\u306f\u4ee5\u4e0b\u306e\u3088\u3046\u306b\uff1a<\/p>\n
$\\displaystyle \\int_{y_1}^{y_2} \\left\\{\\int_{x_1}^{x_2} f(x, y)\\, dx \\right\\} dy = $
\nintegrate(f(x, y), (x, x1, x2), (y, y1, y2))<\/code><\/p>\n<\/div>\n<\/div>\n<\/div>\n
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\u25cb\u7df4\u7fd2\uff1a\u7d2f\u6b21\u7a4d\u5206\u306b\u3088\u308b2\u91cd\u7a4d\u5206<\/h4>\n
$$ I = \\iint_D dx dy, \\quad D: x^2 + y^2 \\leq 1$$<\/p>\n
\u7d2f\u6b21\u7a4d\u5206\u306e\u5f62\u306b\u3059\u308b\u3002<\/p>\n
\u9818\u57df $D$ \u306e\u5b9a\u7fa9\u5f0f\u3092 $y$ \u306b\u3064\u3044\u3066\u89e3\u304d\uff0c<\/p>\n
$$-\\sqrt{1-x^2} \\leq y \\leq \\sqrt{1-x^2}$$<\/p>\n
\u5e73\u65b9\u6839\u306e\u4e2d\u304c\u30bc\u30ed\u4ee5\u4e0a\u3067\u3042\u308b\u3053\u3068\u304b\u3089<\/p>\n
$$ -1 < x < 1$$<\/p>\n
\u88ab\u7a4d\u5206\u95a2\u6570\u306f $f(x, y) = 1$\u3002<\/p>\n
\u30b1\u30fc\u30b9 2 \u306e\u7d2f\u6b21\u7a4d\u5206\u306e\u5f62\uff1a
\n\\begin{eqnarray}
\nf(x, y) &=& 1 \\\\
\nx_1 &=& -1\\\\
\nx_2 &=& 1 \\\\
\ny_1(x) &=& -\\sqrt{1-x^2}\\\\
\ny_2(x) &=& \\sqrt{1-x^2}
\n\\end{eqnarray}<\/p>\n
\\begin{eqnarray}
\n\\iint_D f(x, y)\\,dx dy &=& \\int_{x_1}^{x_2} \\left\\{\\int_{y_1(x)}^{y_2(x) } f(x, y)\\,dy \\right\\} dx \\\\
\n&=& \\cdots
\n\\end{eqnarray}<\/p>\n<\/div>\n<\/div>\n<\/div>\n
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In\u00a0[47]:<\/div>\n
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f<\/span> =<\/span> 1<\/span>\r\nx1<\/span> =<\/span> -<\/span>1<\/span>\r\nx2<\/span> =<\/span> 1<\/span>\r\ny1<\/span> =<\/span> -<\/span>sqrt<\/span>(<\/span>1<\/span>-<\/span>x<\/span>**<\/span>2<\/span>)<\/span>\r\ny2<\/span> =<\/span> sqrt<\/span>(<\/span>1<\/span>-<\/span>x<\/span>**<\/span>2<\/span>)<\/span>\r\n\r\nIntegral<\/span>(<\/span>f<\/span>,<\/span> (<\/span>y<\/span>,<\/span> y1<\/span>,<\/span> y2<\/span>),<\/span> (<\/span>x<\/span>,<\/span> x1<\/span>,<\/span> x2<\/span>))<\/span>\r\n<\/pre>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n
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Out[47]:<\/div>\n
$\\displaystyle \\int\\limits_{-1}^{1}\\int\\limits_{- \\sqrt{1 – x^{2}}}^{\\sqrt{1 – x^{2}}} 1\\, dy\\, dx$<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n
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\u78ba\u8a8d\u306e\u305f\u3081\u306b\uff0c\u6bb5\u968e\u3054\u3068\u306b\u8a08\u7b97\u3057\u3066\u307f\u307e\u3059\u3002<\/p>\n
\u307e\u305a\uff0c$\\displaystyle \\int_{-\\sqrt{1-x^2}}^{\\sqrt{1-x^2}} f \\, dy =$ integrate(f, (y, y1, y2))<\/code><\/p>\n<\/div>\n<\/div>\n<\/div>\n
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In\u00a0[48]:<\/div>\n
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Eq<\/span>(<\/span>Integral<\/span>(<\/span>f<\/span>,<\/span> (<\/span>y<\/span>,<\/span> y1<\/span>,<\/span> y2<\/span>)),<\/span> \r\n   integrate<\/span>(<\/span>f<\/span>,<\/span> (<\/span>y<\/span>,<\/span> y1<\/span>,<\/span> y2<\/span>)))<\/span>\r\n<\/pre>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n
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Out[48]:<\/div>\n
$\\displaystyle \\int\\limits_{- \\sqrt{1 – x^{2}}}^{\\sqrt{1 – x^{2}}} 1\\, dy = 2 \\sqrt{1 – x^{2}}$<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n
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\u6b21\u306b\uff0c$\\displaystyle \\int_{-1}^1 2 \\sqrt{1-x^2}\\, dx$ \u3092\u8a08\u7b97\u3057\u307e\u3059\u3002<\/p>\n
\u4eba\u529b\u3067\u3053\u306e\u7a4d\u5206\u3092\u8a08\u7b97\u3059\u308b\u306b\u306f\uff0c\u4ee5\u4e0b\u306e\u4e0d\u5b9a\u7a4d\u5206\u306e\u9805\u3092\u53c2\u7167\u306e\u3053\u3068\u3002<\/p>\n