{ "cells": [ { "cell_type": "markdown", "metadata": { "toc": "true" }, "source": [ "# Table of Contents\n", "

1  積分(int)
1.1  単純な積分(int)
1.2  Integralいろいろ
1.2.1  部分積分(integration by parts)
1.3  置換(change of variables)による積分
1.3.0.1  部分分数(partial fraction)展開による積分
1.4  課題
1.4.1  不定積分
1.4.2  定積分:次の定積分を求めよ．
1.4.3  (発展課題，重積分）次の2重積分を求めよ．
1.4.4  解答例
" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "
\n", "\n", "
\n", "積分(integral)\n", "
\n", "
\n", "
\n", "file:/~/python/doing_math_with_python/differential.ipynb\n", "
\n", "cc by Shigeto R. Nishitani 2017 \n", "
\n" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "# 積分(int)\n" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "## 単純な積分(int) \n" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "単純な不定積分．\n", "$$\n", "\\int \\ln x\\, dx = x \\ln x -x\n", "$$" ] }, { "cell_type": "code", "execution_count": 3, "metadata": {}, "outputs": [ { "data": { "text/plain": [ "x*log(x) - x" ] }, "execution_count": 3, "metadata": {}, "output_type": "execute_result" } ], "source": [ "from sympy import *\n", "x,y = symbols('x y')\n", "\n", "integrate(ln(x),x)" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "定積分を実行するには，積分変数の範囲を指定する．\n", "$$\n", "\\int_{-\\pi}^0 \\sin x \\, dx = -2\n", "$$" ] }, { "cell_type": "code", "execution_count": 4, "metadata": {}, "outputs": [ { "data": { "text/plain": [ "-2" ] }, "execution_count": 4, "metadata": {}, "output_type": "execute_result" } ], "source": [ "integrate(sin(x),(x,-pi,0))" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "特異点をもつ場合にも適切に積分結果を求めてくれ...\n", "\n", "$$\n", "\\int_0^2 \\frac{1}{\\sqrt{x(2-x)}} dx = \\pi\n", "$$\n" ] }, { "cell_type": "code", "execution_count": 8, "metadata": {}, "outputs": [ { "data": { "text/plain": [ "Integral(1/sqrt(x*(-x + 2)), (x, 0, 2))" ] }, "execution_count": 8, "metadata": {}, "output_type": "execute_result" } ], "source": [ "integrate(1/sqrt(x*(2-x)),(x,0,2))" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "だめみたいですね．どこで引っかかっているかは不明．\n" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "無限区間(infinity)における定積分は計算してくれる．\n", "$$\n", "\\int_{-\\infty}^{\\infty} \\frac{1}{x^2+4} dx = \\frac{\\pi}{2}\n", "$$\n" ] }, { "cell_type": "code", "execution_count": 4, "metadata": {}, "outputs": [ { "data": { "text/plain": [ "pi/2" ] }, "execution_count": 4, "metadata": {}, "output_type": "execute_result" } ], "source": [ "integrate(1/(x**2+4),(x, -oo, oo))" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "部分積分法や置換積分法を用いる必要のある複雑な積分も一発で求まる．\n", "$$\n", "{\\rm eq}\\, = \\, \\sqrt{4-{x}^{2}} \\notag \\\\\n", "\\int {\\rm eq}\\, dx = \\frac{1}{2}\\,x \\sqrt{4-{x}^{2}}+2\\,\\arcsin \\left( 1/2\\,x \\right) \\notag\n", "$$" ] }, { "cell_type": "code", "execution_count": 5, "metadata": {}, "outputs": [ { "data": { "text/plain": [ "x*sqrt(-x**2 + 4)/2 + 2*asin(x/2)" ] }, "execution_count": 5, "metadata": {}, "output_type": "execute_result" } ], "source": [ "eq=sqrt(4-x**2)\n", "integrate(eq,x)" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "数学の公式集を見ないとわからないような積分も同じコマンドで求まる．\n", "$$\n", "{\\rm eq2}\\, = \\,\\exp({-{x}^{2}}) \\notag \\\\\n", "\\int_0^{zz} {\\rm eq}\\, dx = \\frac{1}{2}\\, \\sqrt{\\pi }\\, \\mbox{erf} \\left(zz\\right) \\notag\n", "$$" ] }, { "cell_type": "code", "execution_count": 9, "metadata": {}, "outputs": [ { "data": { "text/plain": [ "sqrt(pi)*erf(zz)/2" ] }, "execution_count": 9, "metadata": {}, "output_type": "execute_result" } ], "source": [ "zz = symbols('zz')\n", "eq2=exp(-x**2)\n", "integrate(eq2, (x,0,zz))" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "## Integralいろいろ" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "ちょっとぐらい難しい積分も，sympyは単純にintegrateコマンドだけで実行してくれます．\n", "しかし，時には，途中の計算法である部分積分，置換積分，部分分数展開が必要になかもしれません．\n", "これらはIntegralというパッケージに幾つか用意されているようです．詳しくは[サイト](http://docs.sympy.org/latest/modules/integrals/integrals.html)を参照してください．\n", "\n", "残念ながら部分積分(integration by parts)の途中を見るコマンドはなさそう．" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "### 部分積分(integration by parts) " ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "$$\n", "\\int x \\exp(x)\\, dx = x\\exp(x)-\\int \\exp(x)\\, {dx}\n", "$$\n", "\n", "mapleでは，\n", "```maple\n", "> intparts(Int(x*exp(x),x),x);\n", "```\n", "\n", "sympyでは，こいつはなさそう．" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "## 置換(change of variables)による積分 " ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "コマンドtransformを使う．\n", "$$\n", "\\int \\left( \\cos \\left( x \\right) +1 \\right) ^{3}\\sin \\left( x \\right) {dx} \\notag\n", "$$\n", "これを$cos(x)+1=u$として置換すると．\n", "$$\n", "\\int-{u}^{3}{du}\n", "$$\n", "となる．\n", "\n", "```maple\n", "> Int((cos(x)+1)^3*sin(x), x);\n", "> changevar(cos(x)+1=u, Int((cos(x)+1)^3*sin(x), x=a..b), u);\n", "> changevar(cos(x)+1=u, int((cos(x)+1)^3*sin(x), x), u);\n", "```\n", "$$\n", "\\int \\left( \\cos \\left( x \\right) +1 \\right) ^{3}\\sin \\left( x \\right) {dx} \\notag \\\\\n", "\\int _{\\cos \\left( a \\right) +1}^{\\cos \\left( b \\right) +1}-{u}^{3}{du} \\notag \\\\\n", "-\\frac{1}{4}\\,{u}^{4} \\notag \\\\\n", "$$\n", "\n" ] }, { "cell_type": "code", "execution_count": 37, "metadata": {}, "outputs": [ { "name": "stdout", "output_type": "stream", "text": [ "⌠ \n", "⎮ 3 \n", "⎮ -u du\n", "⌡ \n" ] } ], "source": [ "a,b,u = symbols('a b u')\n", "\n", "i = Integral((cos(x)+1)**3*sin(x),x)\n", "pprint(i.transform(cos(x)+1,u))" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "定積分(x=1..2)を実行すると，以下のようになる．ただし，$-u^3 \\rightarrow u^3$は積分範囲が反対向けになっているのに注意．" ] }, { "cell_type": "code", "execution_count": 39, "metadata": {}, "outputs": [ { "name": "stdout", "output_type": "stream", "text": [ "cos(1) + 1 \n", " ⌠ \n", " ⎮ 3 \n", " ⎮ u du\n", " ⌡ \n", "cos(2) + 1 \n" ] }, { "data": { "text/plain": [ "-(cos(2) + 1)**4/4 + (cos(1) + 1)**4/4" ] }, "execution_count": 39, "metadata": {}, "output_type": "execute_result" } ], "source": [ "i = Integral((cos(x)+1)**3*sin(x),(x,1,2))\n", "pprint(i.transform(cos(x)+1,u))\n", "integrate(u**3, (u, cos(2) + 1, cos(1) + 1))" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "#### 部分分数(partial fraction)展開による積分 \n" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "部分分数(partial fraction)展開による積分では，apartコマンドを用いる．\n", "$$\n", "{\\rm pf1}\\, = \\frac{1}{1+x^3} = \\frac{1}{3}\\,{\\frac {-x+2}{{x}^{2}-x+1}}+ \\frac{1}{3\\,\\left( x+1 \\right) } \\notag \\\\\n", "\\int {\\rm pf1} dx = -\\frac{1}{6}\\,\\ln \\left( {x}^{2}-x+1 \\right) +\\frac{1}{3}\\, \\sqrt{3}\\arctan \\left( 1/3\\, \\left( 2\\,x-1 \\right) \\sqrt{3} \\right) +\\frac{1}{3}\\,\\ln \\left( x+1 \\right) \\notag\n", "$$" ] }, { "cell_type": "code", "execution_count": 11, "metadata": { "scrolled": true }, "outputs": [ { "name": "stdout", "output_type": "stream", "text": [ " x - 2 1 \n", "- ────────────── + ─────────\n", " ⎛ 2 ⎞ 3⋅(x + 1)\n", " 3⋅⎝x - x + 1⎠ \n", " ⎛2⋅√3⋅x √3⎞\n", " ⎛ 2 ⎞ √3⋅atan⎜────── - ──⎟\n", "log(x + 1) log⎝x - x + 1⎠ ⎝ 3 3 ⎠\n", "────────── - ─────────────── + ────────────────────\n", " 3 6 3 \n" ] } ], "source": [ "eq = 1/(1+x**3)\n", "pprint(apart(eq))\n", "\n", "pprint(integrate(apart(eq),x))" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "## 課題" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "### 不定積分\n", "\n", "次の不定積分を求めよ．\n", "1. $\\int 4\\,x+3{dx}$\n", "1. $ \\int \\frac{1}{ 1+\\mbox{e}^{x} }{dx}$\n", "1. $ \\int \\frac{1}{ \\mbox{e}^{-x}+\\mbox{e}^{x} }{dx}$\n", "1. $ \\int \\sqrt{1-{x}^{2}}{dx}$" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "### 定積分:次の定積分を求めよ．\n", "\n", "1. $ \\int _{0}^{\\pi } \\sin x{dx}$\n", "1. $ \\int _{0}^{1} \\arctan x{dx}$\n", "1. $ \\int _{-2}^{2} \\frac{1}{ \\sqrt{4-{x}^{2}} }{dx}$\n", "1. $ \\int _{0}^{1} \\frac{1}{ {x}^{2}+x+1 }{dx}$" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "### (発展課題，重積分）次の2重積分を求めよ．\n", "\n", "$$\n", "\\int \\int_{D} \\sqrt{x^2+y^2}dxdy\\,\\, D:0\\leq y \\leq x \\leq 1\n", "$$" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "### 解答例" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "1. \n", "```maple\n", "> int(4*x+3,x);\n", "> int( 1/(1+exp(x)),x);\n", "> int(1/(exp(-x)+exp(x)),x);\n", "> int(sqrt(1-x^2),x);\n", "```\n", "$$\n", "2\\,{x}^{2}+3\\,x \\notag \\\\\n", "-\\ln \\left( 1+ \\mbox{e}^x \\right) +\\ln \\left( \\mbox{e}^x \\right) \\notag \\\\\n", "\\arctan \\left( \\mbox{e}^x \\right) \\notag \\\\\n", "\\frac{1}{2}\\,x \\sqrt{1-{x}^{2}}+\\frac{1}{2}\\,\\arcsin \\left( x \\right) \\notag\n", "$$\n", "\n" ] }, { "cell_type": "code", "execution_count": 67, "metadata": {}, "outputs": [ { "name": "stdout", "output_type": "stream", "text": [ " 2 \n", "2⋅x + 3⋅x\n", " ⎛ x ⎞\n", "x - log⎝ℯ + 1⎠\n", " 1 \n", "────────\n", " x -x\n", "ℯ + ℯ \n", " ⎛ 2 ⎛ x⎞⎞\n", "RootSum⎝4⋅z + 1, i ↦ i⋅log⎝2⋅i + ℯ ⎠⎠\n", " __________ \n", " ╱ 2 \n", "x⋅╲╱ - x + 1 asin(x)\n", "─────────────── + ───────\n", " 2 2 \n" ] } ], "source": [ "from sympy import *\n", "x = symbols('x')\n", "\n", "pprint(integrate(4*x+3,x))\n", "pprint(integrate(1/(1+exp(x)),x))\n", "pprint(1/(exp(-x)+exp(x)))\n", "pprint(integrate(1/(exp(-x)+exp(x)),x))\n", "pprint(integrate(sqrt(1-x**2),x))" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "1. \n", "```maple\n", "> int(sin(x),x=0..Pi);\n", "> int(arctan(x),x=0..1);\n", "> int(1/(sqrt(4-x^(2))),x=-2..2);\n", "> int(1/(x^2+x+1),x=0..1);\n", "```\n", "$$\n", "2 \\notag \\\\\n", "\\frac{1}{4}\\,\\pi -\\frac{1}{2}\\,\\ln \\left( 2 \\right) \\notag \\\\\n", "\\pi \\notag \\\\\n", "\\frac{1}{9}\\,\\pi \\, \\sqrt{3} \\notag\n", "$$" ] }, { "cell_type": "code", "execution_count": 81, "metadata": {}, "outputs": [ { "name": "stdout", "output_type": "stream", "text": [ "2\n", " log(2) π\n", "- ────── + ─\n", " 2 4\n", "π\n", "√3⋅π\n", "────\n", " 9 \n" ] } ], "source": [ "pprint(integrate(sin(x),(x,0,pi)))\n", "pprint(integrate(atan(x),(x,0,1)))\n", "pprint(integrate(1/sqrt(4-x**2),(x,-2,2)))\n", "pprint(integrate(1/(x**2+x+1),(x,0,1)))" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "こいつの表示はまだわからん．\n", "```maple\n", "> with(plots):\n", "> inequal({x-y>=0,x>=0,x<=1,y>=0},x=-0.5..1.5,y=-0.5..1.5,optionsexcluded=(color=white));\n", "```\n", "\n", "![](./figs/Intplot2d1.png)" ] }, { "cell_type": "code", "execution_count": 10, "metadata": { "collapsed": true }, "outputs": [], "source": [ "f = sqrt(x**2+y**2)" ] }, { "cell_type": "code", "execution_count": 11, "metadata": {}, "outputs": [ { "data": { "text/plain": [ "log(1 + sqrt(2))/6 + sqrt(2)/6" ] }, "execution_count": 11, "metadata": {}, "output_type": "execute_result" } ], "source": [ "integrate(integrate(f,(y,0,x)),(x,0,1))" ] }, { "cell_type": "code", "execution_count": 3, "metadata": {}, "outputs": [ { "data": { "image/png": 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"text/plain": [ "" ] }, "metadata": {}, "output_type": "display_data" } ], "source": [ "# example in https://stackoverflow.com/questions/17576508/python-matplotlib-drawing-linear-inequality-functions\n", "import numpy as np\n", "import matplotlib.pyplot as plt\n", "from sympy.solvers import solve\n", "from sympy import Symbol\n", "\n", "def f1(x):\n", " return 4.0*x-2.0\n", "def f2(x):\n", " return 0.5*x+2.0\n", "def f3(x):\n", " return -0.3*x+7.0\n", "\n", "x = Symbol('x')\n", "x1, = solve(f1(x)-f2(x))\n", "x2, = solve(f1(x)-f3(x))\n", "x3, = solve(f2(x)-f3(x))\n", "\n", "y1 = f1(x1)\n", "y2 = f1(x2)\n", "y3 = f2(x3)\n", "\n", "plt.plot(x1,f1(x1),'go',markersize=10)\n", "plt.plot(x2,f1(x2),'go',markersize=10)\n", "plt.plot(x3,f2(x3),'go',markersize=10)\n", "\n", "plt.fill([x1,x2,x3,x1],[y1,y2,y3,y1],'red',alpha=0.8)\n", "\n", "xr = np.linspace(0.5,7.5,100)\n", "y1r = f1(xr)\n", "y2r = f2(xr)\n", "y3r = f3(xr)\n", "\n", "plt.plot(xr,y1r,'k--')\n", "plt.plot(xr,y2r,'k--')\n", "plt.plot(xr,y3r,'k--')\n", "\n", "plt.xlim(0.5,7)\n", "plt.ylim(2,8)\n", "\n", "plt.show()" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "fillplotsというのがあるようだが，python3で動かすとエラーが出る．" ] }, { "cell_type": "code", "execution_count": null, "metadata": { "collapsed": true }, "outputs": [], "source": [] } ], "metadata": { "kernelspec": { "display_name": "Python 3", "language": "python", "name": "python3" }, "language_info": { "codemirror_mode": { "name": "ipython", "version": 3 }, "file_extension": ".py", "mimetype": "text/x-python", "name": "python", "nbconvert_exporter": "python", "pygments_lexer": "ipython3", "version": "3.6.1" }, "latex_envs": { "LaTeX_envs_menu_present": true, "autocomplete": true, "bibliofile": "biblio.bib", "cite_by": "apalike", "current_citInitial": 1, "eqLabelWithNumbers": true, "eqNumInitial": 1, "hotkeys": { "equation": "Ctrl-E", "itemize": "Ctrl-I" }, "labels_anchors": false, "latex_user_defs": false, "report_style_numbering": false, "user_envs_cfg": false }, "toc": { "colors": { "hover_highlight": "#DAA520", "navigate_num": "#000000", "navigate_text": "#333333", "running_highlight": "#FF0000", "selected_highlight": "#FFD700", "sidebar_border": "#EEEEEE", "wrapper_background": "#FFFFFF" }, "moveMenuLeft": true, "nav_menu": { "height": "12px", "width": "252px" }, "navigate_menu": true, "number_sections": true, "sideBar": true, "threshold": 4, "toc_cell": true, "toc_section_display": "block", "toc_window_display": false, "widenNotebook": false } }, "nbformat": 4, "nbformat_minor": 2 }