MathWithSumpy_INTEGRALS_082918.ipynb

{
 "cells": [
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Just about any Integral !!!\n",
    "###  Definite and Indefinite\n",
    " \n",
    " - Derivative of one dimensional variable\n",
    " - Derivative of multimensional variable\n",
    " \n",
    " #### If integrate function is unable to compute an integral, it returns an unevaluated Integral object\n"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 2,
   "metadata": {
    "collapsed": true
   },
   "outputs": [],
   "source": [
    "from sympy import *\n",
    "init_printing()"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 3,
   "metadata": {
    "collapsed": false
   },
   "outputs": [],
   "source": [
    "x, y, z = symbols(\"x, y, z\")"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 4,
   "metadata": {
    "collapsed": true
   },
   "outputs": [],
   "source": [
    "a, b,c,d = symbols ('a b c d')"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### Simple One Dimension  -- Indefinite Integral"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 5,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "image/png": "iVBORw0KGgoAAAANSUhEUgAAAEYAAAAuBAMAAABnp3KAAAAAMFBMVEX///8AAAAAAAAAAAAAAAAA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\nSuZ0WbnVDw3qYORVuv/1AAAAAElFTkSuQmCC\n",
      "text/latex": [
       "$$\\int x^{3}\\, dx$$"
      ],
      "text/plain": [
       "⌠      \n",
       "⎮  3   \n",
       "⎮ x  dx\n",
       "⌡      "
      ]
     },
     "execution_count": 5,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "Integral ( x **3 , x )"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 6,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "image/png": "iVBORw0KGgoAAAANSUhEUgAAABYAAAAwBAMAAADnb5cjAAAAMFBMVEX///8AAAAAAAAAAAAAAAAA\nAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAv3aB7AAAAD3RSTlMAEHarIkSJZt3NVLsy\nme8Q6PJIAAAACXBIWXMAAA7EAAAOxAGVKw4bAAAAsUlEQVQoFWNggAJuGANIrxeAcxjzEWwmeQT7\nFBI7AcFmd0CwhRgQ7LV371+Hm8ngDzJHSNlVTYGBfb8VAwNjAnsTxwSIPJsA20cuBwibkYFzA4QF\nIvkVEOzzB2BsLgF9Biaou+IX7Gdwh0oIKwldL4ApojH9HwGoYNNZJDO6EWym1wi2+2wEOwHBZndA\nsIUYEOwEBJvrAIItxoBgLzY2/meEMBTJLobfCOHs/+YIDpgFAABhK9CCxfYhAAAAAElFTkSuQmCC\n",
      "text/latex": [
       "$$\\frac{x^{4}}{4}$$"
      ],
      "text/plain": [
       " 4\n",
       "x \n",
       "──\n",
       "4 "
      ]
     },
     "execution_count": 6,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "integrate ( x **3 , x )"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 7,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "image/png": "iVBORw0KGgoAAAANSUhEUgAAAGsAAAAuBAMAAADdBPrgAAAAMFBMVEX///8AAAAAAAAAAAAAAAAA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\nuq90X5nb1zPE8GTaS4//AWZcmEkaZZE7AAAAAElFTkSuQmCC\n",
      "text/latex": [
       "$$\\int \\log{\\left (x \\right )}\\, dx$$"
      ],
      "text/plain": [
       "⌠          \n",
       "⎮ log(x) dx\n",
       "⌡          "
      ]
     },
     "execution_count": 7,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    " Integral ( ln (x ), x )"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 8,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "image/png": "iVBORw0KGgoAAAANSUhEUgAAAHMAAAAuBAMAAADpfTv8AAAAMFBMVEX///8AAAAAAAAAAAAAAAAA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\n",
      "text/latex": [
       "$$\\int \\sin^{2}{\\left (x \\right )}\\, dx$$"
      ],
      "text/plain": [
       "⌠           \n",
       "⎮    2      \n",
       "⎮ sin (x) dx\n",
       "⌡           "
      ]
     },
     "execution_count": 8,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "Integral(sin(x)**2)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 23,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "image/png": "iVBORw0KGgoAAAANSUhEUgAAALAAAAAqBAMAAADhZMxIAAAAMFBMVEX///8AAAAAAAAAAAAAAAAA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\nzfzZljkJAAAAAElFTkSuQmCC\n",
      "text/latex": [
       "$$\\frac{x}{2} - \\frac{1}{2} \\sin{\\left (x \\right )} \\cos{\\left (x \\right )}$$"
      ],
      "text/plain": [
       "x   sin(x)⋅cos(x)\n",
       "─ - ─────────────\n",
       "2         2      "
      ]
     },
     "execution_count": 23,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "integrate(sin(x)**2)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 9,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "image/png": "iVBORw0KGgoAAAANSUhEUgAAAG4AAAAUBAMAAACezBVvAAAAMFBMVEX///8AAAAAAAAAAAAAAAAA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\nnuE8YhjVGwijOTb/T/wAtM1kFi/cfW4AAAAASUVORK5CYII=\n",
      "text/latex": [
       "$$x \\log{\\left (x \\right )} - x$$"
      ],
      "text/plain": [
       "x⋅log(x) - x"
      ]
     },
     "execution_count": 9,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    " integrate ( ln (x ), x )"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### Simple One Dimension  -- Definite Integral\n",
    "#### Note that complex integrals take a while to run"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 10,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "image/png": "iVBORw0KGgoAAAANSUhEUgAAAEYAAABKBAMAAAD0wj/MAAAAMFBMVEX///8AAAAAAAAAAAAAAAAA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\nYII=\n",
      "text/latex": [
       "$$\\int_{0}^{1} x^{3}\\, dx$$"
      ],
      "text/plain": [
       "1      \n",
       "⌠      \n",
       "⎮  3   \n",
       "⎮ x  dx\n",
       "⌡      \n",
       "0      "
      ]
     },
     "execution_count": 10,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    " Integral ( x **3 , (x , 0 , 1))"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 11,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "image/png": "iVBORw0KGgoAAAANSUhEUgAAAAsAAAArBAMAAABcLm8jAAAALVBMVEX///8AAAAAAAAAAAAAAAAA\nAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAADAOrOgAAAADnRSTlMAVO8Qq5l2zWYy3bsiiSz4\nAk0AAAAJcEhZcwAADsQAAA7EAZUrDhsAAABcSURBVBgZY2CAASEDIIvZ1Q9EMTDkUYMK0StpAJtG\nFvEOBB4QqXUqRN1qMMWmDaZOVIOpC2CK4wCYMmMAUxfAFOMEMGXLAKZEXFweuYF1QLQzvAZzbr/z\nANNAAgAV7BweLz/3QAAAAABJRU5ErkJggg==\n",
      "text/latex": [
       "$$\\frac{1}{4}$$"
      ],
      "text/plain": [
       "1/4"
      ]
     },
     "execution_count": 11,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    " integrate ( x **3 , (x , 0 , 1))"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 12,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "image/png": "iVBORw0KGgoAAAANSUhEUgAAAJgAAAAuBAMAAAAmf816AAAAMFBMVEX///8AAAAAAAAAAAAAAAAA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\n",
      "text/latex": [
       "$$\\int x^{2} e^{x} \\cos{\\left (x \\right )}\\, dx$$"
      ],
      "text/plain": [
       "⌠                \n",
       "⎮  2  x          \n",
       "⎮ x ⋅ℯ ⋅cos(x) dx\n",
       "⌡                "
      ]
     },
     "execution_count": 12,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "Integral(x**2 * exp(x) * cos(x), x)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 13,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "image/png": "iVBORw0KGgoAAAANSUhEUgAAAkIAAAAvBAMAAAD9fLieAAAAMFBMVEX///8AAAAAAAAAAAAAAAAA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\n",
      "text/latex": [
       "$$\\frac{x^{2} e^{x}}{2} \\sin{\\left (x \\right )} + \\frac{x^{2} e^{x}}{2} \\cos{\\left (x \\right )} - x e^{x} \\sin{\\left (x \\right )} + \\frac{e^{x}}{2} \\sin{\\left (x \\right )} - \\frac{e^{x}}{2} \\cos{\\left (x \\right )}$$"
      ],
      "text/plain": [
       " 2  x           2  x                         x           x       \n",
       "x ⋅ℯ ⋅sin(x)   x ⋅ℯ ⋅cos(x)      x          ℯ ⋅sin(x)   ℯ ⋅cos(x)\n",
       "──────────── + ──────────── - x⋅ℯ ⋅sin(x) + ───────── - ─────────\n",
       "     2              2                           2           2    "
      ]
     },
     "execution_count": 13,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "integrate(x**2 * exp(x) * cos(x), x)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 14,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "image/png": "iVBORw0KGgoAAAANSUhEUgAAAJYAAAAuBAMAAAA4tv3JAAAAMFBMVEX///8AAAAAAAAAAAAAAAAA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\njBG6rChXHPNanXfsFf8DQQOobmWmo2wAAAAASUVORK5CYII=\n",
      "text/latex": [
       "$$\\int e^{- x^{2}} \\operatorname{erf}{\\left (x \\right )}\\, dx$$"
      ],
      "text/plain": [
       "⌠               \n",
       "⎮    2          \n",
       "⎮  -x           \n",
       "⎮ ℯ   ⋅erf(x) dx\n",
       "⌡               "
      ]
     },
     "execution_count": 14,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "Integral(exp(-x**2)*erf(x),x)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 15,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "image/png": "iVBORw0KGgoAAAANSUhEUgAAAGQAAAAuBAMAAAAsD6FtAAAAMFBMVEX///8AAAAAAAAAAAAAAAAA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\n",
      "text/latex": [
       "$$\\frac{\\sqrt{\\pi}}{4} \\operatorname{erf}^{2}{\\left (x \\right )}$$"
      ],
      "text/plain": [
       "      2   \n",
       "√π⋅erf (x)\n",
       "──────────\n",
       "    4     "
      ]
     },
     "execution_count": 15,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "integrate(exp(-x**2)*erf(x),x)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### Transforms -- Laplace and Fourier"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 16,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "image/png": "iVBORw0KGgoAAAANSUhEUgAAAR4AAAAyBAMAAAB4/qkcAAAAMFBMVEX///8AAAAAAAAAAAAAAAAA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\n5XpmfIb1GMv9L4FLPzWEekOSAAAAAElFTkSuQmCC\n",
      "text/latex": [
       "$$\\left ( \\frac{s^{- a}}{s} \\Gamma{\\left(a + 1 \\right)}, \\quad 0, \\quad - \\Re{a} < 1\\right )$$"
      ],
      "text/plain": [
       "⎛ -a                        ⎞\n",
       "⎜s  ⋅Γ(a + 1)               ⎟\n",
       "⎜────────────, 0, -re(a) < 1⎟\n",
       "⎝     s                     ⎠"
      ]
     },
     "execution_count": 16,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "from sympy.integrals import laplace_transform\n",
    "from sympy.abc import t,s,a\n",
    "laplace_transform(t**a, t, s)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 17,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "image/png": "iVBORw0KGgoAAAANSUhEUgAAAFYAAAAbBAMAAAAUvmV2AAAAMFBMVEX///8AAAAAAAAAAAAAAAAA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\nAElFTkSuQmCC\n",
      "text/latex": [
       "$$\\sqrt{\\pi} e^{- \\pi^{2} k^{2}}$$"
      ],
      "text/plain": [
       "      2  2\n",
       "    -π ⋅k \n",
       "√π⋅ℯ      "
      ]
     },
     "execution_count": 17,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "from sympy import fourier_transform, exp\n",
    "from sympy.abc import x, k\n",
    "fourier_transform(exp(-x**2), x, k)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### Multiple Integrals"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 18,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "image/png": "iVBORw0KGgoAAAANSUhEUgAAALIAAABKBAMAAADtZRMvAAAAMFBMVEX///8AAAAAAAAAAAAAAAAA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\nrky9vPTLyc8698Vhlf8GT2lFWrpTw6IAAAAASUVORK5CYII=\n",
      "text/latex": [
       "$$\\int_{22}^{30}\\int_{20}^{x - 2} x^{2} + y^{2}\\, dy\\, dx$$"
      ],
      "text/plain": [
       "30 x - 2                \n",
       "⌠    ⌠                  \n",
       "⎮    ⎮   ⎛ 2    2⎞      \n",
       "⎮    ⎮   ⎝x  + y ⎠ dy dx\n",
       "⌡    ⌡                  \n",
       "22   20                 "
      ]
     },
     "execution_count": 18,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "f = (x ** 2 + y ** 2)\n",
    "Integral(f, (y, 20, x-2), (x, 22, 30))"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 19,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "image/png": "iVBORw0KGgoAAAANSUhEUgAAAGAAAAAvBAMAAADuuNKyAAAAMFBMVEX///8AAAAAAAAAAAAAAAAA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\n1L8AAAAASUVORK5CYII=\n",
      "text/latex": [
       "$$\\frac{x^{3} y}{3} + \\frac{x y^{3}}{3}$$"
      ],
      "text/plain": [
       " 3        3\n",
       "x ⋅y   x⋅y \n",
       "──── + ────\n",
       " 3      3  "
      ]
     },
     "execution_count": 19,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "integrate(f, y,x)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 20,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "image/png": "iVBORw0KGgoAAAANSUhEUgAAADIAAAAPBAMAAABU9YHxAAAAMFBMVEX///8AAAAAAAAAAAAAAAAA\nAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAv3aB7AAAAD3RSTlMAMpndu3bvImbNiRBU\nq0Qb3U6NAAAACXBIWXMAAA7EAAAOxAGVKw4bAAABJ0lEQVQYGXWPPUvDUBSGn1xjG2Kahv6C6qgg\nHRwViquzg04Z3BQMTm510TlCBBGHCgoSEESx0C7tHxDiUChCoYt/QPyiaRvPrbOHA/ec9zn3fACU\n4OH2Cs7uSx1jbTGOId72BLAClzwG7GdDrCzLxjQHKhQwu4PlY4dcdANUAAnrmFUhhxFmgpOKAra4\n535LJFaPKCa4oymR/A3T1zpWENEu437SO57XQpXi6c2yBE9E1OoYQzaoDaTZEe1zch1ppklZEzAr\n0BZPUbs4npC/bqBk9BYUK1gTWgiRDZxRIcSeTK+bCbF+eO73v7qyvZXmQlTK3BgKvv4jNr1UhQqk\n2HgHQ+b4mnzAKg0vX+YkwBHCK42OPNdZj9bmCxzcLUF+TyR7QdJ/7RcbCFuL1q2iGQAAAABJRU5E\nrkJggg==\n",
      "text/latex": [
       "$$40576$$"
      ],
      "text/plain": [
       "40576"
      ]
     },
     "execution_count": 20,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "integrate(f, (y, 20, x-2), (x, 22, 30))"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 21,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "image/png": "iVBORw0KGgoAAAANSUhEUgAAAO4AAAAuBAMAAADdKDilAAAAMFBMVEX///8AAAAAAAAAAAAAAAAA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\nHf8DcTJFTavV63gAAAAASUVORK5CYII=\n",
      "text/latex": [
       "$$\\iiint x^{2} + y^{2} + z^{2}\\, dy\\, dx\\, dz$$"
      ],
      "text/plain": [
       "⌠ ⌠ ⌠                        \n",
       "⎮ ⎮ ⎮ ⎛ 2    2    2⎞         \n",
       "⎮ ⎮ ⎮ ⎝x  + y  + z ⎠ dy dx dz\n",
       "⌡ ⌡ ⌡                        "
      ]
     },
     "execution_count": 21,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "f = (x ** 2 + y ** 2+ z**2)\n",
    "Integral(f, y,x,z)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 22,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "image/png": "iVBORw0KGgoAAAANSUhEUgAAALcAAAA1BAMAAADxOGd/AAAAMFBMVEX///8AAAAAAAAAAAAAAAAA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\n",
      "text/latex": [
       "$$\\frac{x^{3} y}{3} + x \\left(\\frac{y^{3}}{3} + y z^{2}\\right)$$"
      ],
      "text/plain": [
       " 3       ⎛ 3       ⎞\n",
       "x ⋅y     ⎜y       2⎟\n",
       "──── + x⋅⎜── + y⋅z ⎟\n",
       " 3       ⎝3        ⎠"
      ]
     },
     "execution_count": 22,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "f = (x ** 2 + y ** 2+ z**2)\n",
    "integrate(f, y,x)"
   ]
  }
 ],
 "metadata": {
  "anaconda-cloud": {},
  "kernelspec": {
   "display_name": "Python [conda env:Anaconda3]",
   "language": "python",
   "name": "conda-env-Anaconda3-py"
  },
  "language_info": {
   "codemirror_mode": {
    "name": "ipython",
    "version": 3
   },
   "file_extension": ".py",
   "mimetype": "text/x-python",
   "name": "python",
   "nbconvert_exporter": "python",
   "pygments_lexer": "ipython3",
   "version": "3.5.2"
  }
 },
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}