Lektion02.ipynb

{
 "cells": [
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "# Lektion 2"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 1,
   "metadata": {
    "collapsed": true
   },
   "outputs": [],
   "source": [
    "from sympy import *\n",
    "init_printing()"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Vereinfachungen"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 2,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "image/png": "iVBORw0KGgoAAAANSUhEUgAAAIAAAAAUBAMAAAC35yitAAAAMFBMVEX///8AAAAAAAAAAAAAAAAA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\nzGM49E8AAAAASUVORK5CYII=\n",
      "text/latex": [
       "$$\\left(x - y\\right) \\left(x + y\\right)$$"
      ],
      "text/plain": [
       "(x - y)⋅(x + y)"
      ]
     },
     "execution_count": 2,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "x = Symbol('x')\n",
    "y = Symbol('y')\n",
    "f = (x - y) * (x + y)\n",
    "f"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 3,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "image/png": "iVBORw0KGgoAAAANSUhEUgAAAEIAAAAYBAMAAAC8bzZKAAAAMFBMVEX///8AAAAAAAAAAAAAAAAA\nAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAv3aB7AAAAD3RSTlMAEHarIkSJZt3NVLsy\nme8Q6PJIAAAACXBIWXMAAA7EAAAOxAGVKw4bAAAA/0lEQVQoFWNggIDcvQJQFgqFEGZzYJ6AIgXh\nIAlzOPB8waICSZgjgOUrNhXIwlwfsahgYEAS5lDAqgJJOBerAgaEMPsFrCqQhMMYYrApQQjzzFq1\nEYsKJGG+//8/YFEBERZSdlVTwJQVusTA8hAszJjA3sSBJcQT1jEwQYxlE2D7yOWAYQbLgpsM3Bsg\nZjBwQhggHuOMDiDoPABkMTxi4FcAiQEBnAHhQknWjwznD0DZcAaKCt4JDHoQAS4BfQYmARRJMIfb\ngGEzRDR+wX4Gd0wFDLwbWL9DhIWVhK4XYFHBaKaLLU2hqAS6BC+wZJAPwKuAoZ1hE34FDEG6ByAq\nABw3OaVEK2PEAAAAAElFTkSuQmCC\n",
      "text/latex": [
       "$$x^{2} - y^{2}$$"
      ],
      "text/plain": [
       " 2    2\n",
       "x  - y "
      ]
     },
     "execution_count": 3,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "f.expand()"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 4,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "image/png": "iVBORw0KGgoAAAANSUhEUgAAAEIAAAAYBAMAAAC8bzZKAAAAMFBMVEX///8AAAAAAAAAAAAAAAAA\nAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAv3aB7AAAAD3RSTlMAEHarIkSJZt3NVLsy\nme8Q6PJIAAAACXBIWXMAAA7EAAAOxAGVKw4bAAAA/0lEQVQoFWNggIDcvQJQFgqFEGZzYJ6AIgXh\nIAlzOPB8waICSZgjgOUrNhXIwlwfsahgYEAS5lDAqgJJOBerAgaEMPsFrCqQhMMYYrApQQjzzFq1\nEYsKJGG+//8/YFEBERZSdlVTwJQVusTA8hAszJjA3sSBJcQT1jEwQYxlE2D7yOWAYQbLgpsM3Bsg\nZjBwQhggHuOMDiDoPABkMTxi4FcAiQEBnAHhQknWjwznD0DZcAaKCt4JDHoQAS4BfQYmARRJMIfb\ngGEzRDR+wX4Gd0wFDLwbWL9DhIWVhK4XYFHBaKaLLU2hqAS6BC+wZJAPwKuAoZ1hE34FDEG6ByAq\nABw3OaVEK2PEAAAAAElFTkSuQmCC\n",
      "text/latex": [
       "$$x^{2} - y^{2}$$"
      ],
      "text/plain": [
       " 2    2\n",
       "x  - y "
      ]
     },
     "execution_count": 4,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "expand(f)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 5,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "image/png": "iVBORw0KGgoAAAANSUhEUgAAAI4AAAAYBAMAAADe7NhlAAAAMFBMVEX///8AAAAAAAAAAAAAAAAA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\nqxHsBEqf/TJKvVLvCPo/acuDiMDVIgkAAAAASUVORK5CYII=\n",
      "text/latex": [
       "$$x^{4} - 2 x^{2} y^{2} + y^{4}$$"
      ],
      "text/plain": [
       " 4      2  2    4\n",
       "x  - 2⋅x ⋅y  + y "
      ]
     },
     "execution_count": 5,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "expand(f**2)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 6,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "image/png": "iVBORw0KGgoAAAANSUhEUgAAAEIAAAAYBAMAAAC8bzZKAAAAMFBMVEX///8AAAAAAAAAAAAAAAAA\nAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAv3aB7AAAAD3RSTlMAEHarIkSJZt3NVLsy\nme8Q6PJIAAAACXBIWXMAAA7EAAAOxAGVKw4bAAAA/0lEQVQoFWNggIDcvQJQFgqFEGZzYJ6AIgXh\nIAlzOPB8waICSZgjgOUrNhXIwlwfsahgYEAS5lDAqgJJOBerAgaEMPsFrCqQhMMYYrApQQjzzFq1\nEYsKJGG+//8/YFEBERZSdlVTwJQVusTA8hAszJjA3sSBJcQT1jEwQYxlE2D7yOWAYQbLgpsM3Bsg\nZjBwQhggHuOMDiDoPABkMTxi4FcAiQEBnAHhQknWjwznD0DZcAaKCt4JDHoQAS4BfQYmARRJMIfb\ngGEzRDR+wX4Gd0wFDLwbWL9DhIWVhK4XYFHBaKaLLU2hqAS6BC+wZJAPwKuAoZ1hE34FDEG6ByAq\nABw3OaVEK2PEAAAAAElFTkSuQmCC\n",
      "text/latex": [
       "$$x^{2} - y^{2}$$"
      ],
      "text/plain": [
       " 2    2\n",
       "x  - y "
      ]
     },
     "execution_count": 6,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "fe = f.expand()\n",
    "fe"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 7,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "image/png": "iVBORw0KGgoAAAANSUhEUgAAAIAAAAAUBAMAAAC35yitAAAAMFBMVEX///8AAAAAAAAAAAAAAAAA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\nzGM49E8AAAAASUVORK5CYII=\n",
      "text/latex": [
       "$$\\left(x - y\\right) \\left(x + y\\right)$$"
      ],
      "text/plain": [
       "(x - y)⋅(x + y)"
      ]
     },
     "execution_count": 7,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "fe.factor()"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 8,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "image/png": "iVBORw0KGgoAAAANSUhEUgAAAEoAAAA3BAMAAABZQsFdAAAAMFBMVEX///8AAAAAAAAAAAAAAAAA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\n/KX1AAAAAElFTkSuQmCC\n",
      "text/latex": [
       "$$\\frac{x^{2} - y^{2}}{\\left(x - y\\right)^{2}}$$"
      ],
      "text/plain": [
       " 2    2 \n",
       "x  - y  \n",
       "────────\n",
       "       2\n",
       "(x - y) "
      ]
     },
     "execution_count": 8,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "g = (x**2-y**2)/(x-y)**2\n",
    "g"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 9,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "image/png": "iVBORw0KGgoAAAANSUhEUgAAAFUAAAAuBAMAAACrPkhrAAAAMFBMVEX///8AAAAAAAAAAAAAAAAA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\nSUVORK5CYII=\n",
      "text/latex": [
       "$$\\frac{2 y}{x - y} + 1$$"
      ],
      "text/plain": [
       " 2⋅y     \n",
       "───── + 1\n",
       "x - y    "
      ]
     },
     "execution_count": 9,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "g.ratsimp()"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 10,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "image/png": "iVBORw0KGgoAAAANSUhEUgAAADEAAAAtBAMAAAD1pp7PAAAAMFBMVEX///8AAAAAAAAAAAAAAAAA\nAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAv3aB7AAAAD3RSTlMAEHarIkSJZt3NVLsy\nme8Q6PJIAAAACXBIWXMAAA7EAAAOxAGVKw4bAAABAUlEQVQ4EWNgIAZwKeBSRbyMkLKrGsQYmB6h\nSwwsDxkYGBPYmzgmgC2AySSsY2D6wMDAJsD2kcsBWYZlwU0G7g1APQycQBKoYubMGS9nzjQAiTxi\n4FcAiUFIIANmGutHhvMHQDIQEkmGdwKDHogroM/AJABSAdfDbcCwGciNX7CfwR0sAZfh3cD6HSgi\nrCR0vQBVhtFM9wtEBEbCXMDAALQJBTA6QLiWDPIBKBJwTjvDJjgblRGkewBVgLa8/zjAB9paO1RM\nR6RrmIsx0zVMBjNdQ2XQ0zXjjA4g6DyANV3DTMNM1zAZzHQNk8FM1zAZzHQNk8FM1zAZzHQNkyE/\nXQMATBtba/PkUzoAAAAASUVORK5CYII=\n",
      "text/latex": [
       "$$\\frac{x + y}{x - y}$$"
      ],
      "text/plain": [
       "x + y\n",
       "─────\n",
       "x - y"
      ]
     },
     "execution_count": 10,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "g.factor()"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 11,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "image/png": "iVBORw0KGgoAAAANSUhEUgAAADEAAAAtBAMAAAD1pp7PAAAAMFBMVEX///8AAAAAAAAAAAAAAAAA\nAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAv3aB7AAAAD3RSTlMAEHarIkSJZt3NVLsy\nme8Q6PJIAAAACXBIWXMAAA7EAAAOxAGVKw4bAAABAUlEQVQ4EWNgIAZwKeBSRbyMkLKrGsQYmB6h\nSwwsDxkYGBPYmzgmgC2AySSsY2D6wMDAJsD2kcsBWYZlwU0G7g1APQycQBKoYubMGS9nzjQAiTxi\n4FcAiUFIIANmGutHhvMHQDIQEkmGdwKDHogroM/AJABSAdfDbcCwGciNX7CfwR0sAZfh3cD6HSgi\nrCR0vQBVhtFM9wtEBEbCXMDAALQJBTA6QLiWDPIBKBJwTjvDJjgblRGkewBVgLa8/zjAB9paO1RM\nR6RrmIsx0zVMBjNdQ2XQ0zXjjA4g6DyANV3DTMNM1zAZzHQNk8FM1zAZzHQNk8FM1zAZzHQNkyE/\nXQMATBtba/PkUzoAAAAASUVORK5CYII=\n",
      "text/latex": [
       "$$\\frac{x + y}{x - y}$$"
      ],
      "text/plain": [
       "x + y\n",
       "─────\n",
       "x - y"
      ]
     },
     "execution_count": 11,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "g.simplify()"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 12,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "image/png": "iVBORw0KGgoAAAANSUhEUgAAACEAAAAQBAMAAABq7AtUAAAAMFBMVEX///8AAAAAAAAAAAAAAAAA\nAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAv3aB7AAAAD3RSTlMAEHarIkSJZt3NVLsy\nme8Q6PJIAAAACXBIWXMAAA7EAAAOxAGVKw4bAAAAk0lEQVQYGWNgQAKsyQwcAkh8BgY2foFzKAIM\njhEM2qgiDGYMRmgirQxv0ER6WD+giQRrTwCKCCm7qilASEYH7gIGBsYE9iaOCRCSOSESqIRNgO0j\nlwOEZC1LAIowMnBugJFALhjwK4AoCAkWYDh/AERDSBCLS0CfgUkAQoL4DAzxC/YzuENJiIiwktD1\nAgYICRYBAJczHPGywRA/AAAAAElFTkSuQmCC\n",
      "text/latex": [
       "$$x x^{y}$$"
      ],
      "text/plain": [
       "   y\n",
       "x⋅x "
      ]
     },
     "execution_count": 12,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "h = x*x**y\n",
    "h"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 13,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "image/png": "iVBORw0KGgoAAAANSUhEUgAAACkAAAAUBAMAAADiqgm2AAAAMFBMVEX///8AAAAAAAAAAAAAAAAA\nAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAv3aB7AAAAD3RSTlMAEHarIkSJZt3NVLsy\nme8Q6PJIAAAACXBIWXMAAA7EAAAOxAGVKw4bAAAAtklEQVQYGWNgwAA8EzCEGBh4LjegiTKD+Fzo\notyooqzJDBwCDAxoomz8AueA6tBEHSMYtDFFGcwYjBgsZ86bOVMB2bZWhjdY1PawfoCLPgCyICBY\nG+QjkG0sNz4rMDAIKbuqKTA6cBdARYEUAwNjAnsTxwTmhEgQB+wyEINNgO0jlwNrWQKIwwIiQICR\ngXMDhIVK8iug8iG88wcwRbkE9BmYBNDF4xfsZ3BHF2QQVhK6DnIqCgAAlVUeuLlUNuMAAAAASUVO\nRK5CYII=\n",
      "text/latex": [
       "$$x^{y + 1}$$"
      ],
      "text/plain": [
       " y + 1\n",
       "x     "
      ]
     },
     "execution_count": 13,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "h.powsimp()"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 14,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "image/png": "iVBORw0KGgoAAAANSUhEUgAAACkAAAAUBAMAAADiqgm2AAAAMFBMVEX///8AAAAAAAAAAAAAAAAA\nAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAv3aB7AAAAD3RSTlMAEHarIkSJZt3NVLsy\nme8Q6PJIAAAACXBIWXMAAA7EAAAOxAGVKw4bAAAAtklEQVQYGWNgwAA8EzCEGBh4LjegiTKD+Fzo\notyooqzJDBwCDAxoomz8AueA6tBEHSMYtDFFGcwYjBgsZ86bOVMB2bZWhjdY1PawfoCLPgCyICBY\nG+QjkG0sNz4rMDAIKbuqKTA6cBdARYEUAwNjAnsTxwTmhEgQB+wyEINNgO0jlwNrWQKIwwIiQICR\ngXMDhIVK8iug8iG88wcwRbkE9BmYBNDF4xfsZ3BHF2QQVhK6DnIqCgAAlVUeuLlUNuMAAAAASUVO\nRK5CYII=\n",
      "text/latex": [
       "$$x^{y + 1}$$"
      ],
      "text/plain": [
       " y + 1\n",
       "x     "
      ]
     },
     "execution_count": 14,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "hp = h.simplify()\n",
    "hp"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 15,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "image/png": "iVBORw0KGgoAAAANSUhEUgAAACEAAAAQBAMAAABq7AtUAAAAMFBMVEX///8AAAAAAAAAAAAAAAAA\nAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAv3aB7AAAAD3RSTlMAEHarIkSJZt3NVLsy\nme8Q6PJIAAAACXBIWXMAAA7EAAAOxAGVKw4bAAAAk0lEQVQYGWNgQAKsyQwcAkh8BgY2foFzKAIM\njhEM2qgiDGYMRmgirQxv0ER6WD+giQRrTwCKCCm7qilASEYH7gIGBsYE9iaOCRCSOSESqIRNgO0j\nlwOEZC1LAIowMnBugJFALhjwK4AoCAkWYDh/AERDSBCLS0CfgUkAQoL4DAzxC/YzuENJiIiwktD1\nAgYICRYBAJczHPGywRA/AAAAAElFTkSuQmCC\n",
      "text/latex": [
       "$$x x^{y}$$"
      ],
      "text/plain": [
       "   y\n",
       "x⋅x "
      ]
     },
     "execution_count": 15,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "hp.expand()"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 16,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "image/png": "iVBORw0KGgoAAAANSUhEUgAAAWoAAAA1BAMAAACaWbxGAAAAMFBMVEX///8AAAAAAAAAAAAAAAAA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\nqEYeXPcxEoPXHZIBO50JgY0C9B94E+jlJwit+gAAAABJRU5ErkJggg==\n",
      "text/latex": [
       "$$\\frac{\\sin{\\left (2 x \\right )} + \\cos{\\left (x \\right )}}{\\left(\\sin{\\left (2 x \\right )} - \\cos{\\left (x \\right )}\\right) \\left(\\sin^{2}{\\left (2 x \\right )} - \\cos^{2}{\\left (x \\right )}\\right)}$$"
      ],
      "text/plain": [
       "            sin(2⋅x) + cos(x)            \n",
       "─────────────────────────────────────────\n",
       "                    ⎛   2           2   ⎞\n",
       "(sin(2⋅x) - cos(x))⋅⎝sin (2⋅x) - cos (x)⎠"
      ]
     },
     "execution_count": 16,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "f = (sin(2*x)+cos(x)) / ((sin(2*x)**2 - cos(x)**2)*(sin(2*x)-cos(x)))\n",
    "f"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 17,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "image/png": "iVBORw0KGgoAAAANSUhEUgAAAVQAAAAxBAMAAAB38kzbAAAAMFBMVEX///8AAAAAAAAAAAAAAAAA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\nrkJggg==\n",
      "text/latex": [
       "$$\\frac{1}{\\sin^{2}{\\left (2 x \\right )} - 2 \\sin{\\left (2 x \\right )} \\cos{\\left (x \\right )} + \\cos^{2}{\\left (x \\right )}}$$"
      ],
      "text/plain": [
       "                   1                   \n",
       "───────────────────────────────────────\n",
       "   2                               2   \n",
       "sin (2⋅x) - 2⋅sin(2⋅x)⋅cos(x) + cos (x)"
      ]
     },
     "execution_count": 17,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "f.ratsimp()"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 18,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "image/png": "iVBORw0KGgoAAAANSUhEUgAAAM0AAAAzBAMAAADGCpEzAAAAMFBMVEX///8AAAAAAAAAAAAAAAAA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\nwn/53fZdw8jzld/tPzHAXRIN8AjDAAAAAElFTkSuQmCC\n",
      "text/latex": [
       "$$\\frac{1}{\\left(2 \\sin{\\left (x \\right )} - 1\\right)^{2} \\cos^{2}{\\left (x \\right )}}$$"
      ],
      "text/plain": [
       "           1           \n",
       "───────────────────────\n",
       "              2    2   \n",
       "(2⋅sin(x) - 1) ⋅cos (x)"
      ]
     },
     "execution_count": 18,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "f.trigsimp()"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 19,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "image/png": "iVBORw0KGgoAAAANSUhEUgAAAM0AAAAzBAMAAADGCpEzAAAAMFBMVEX///8AAAAAAAAAAAAAAAAA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\nwn/53fZdw8jzld/tPzHAXRIN8AjDAAAAAElFTkSuQmCC\n",
      "text/latex": [
       "$$\\frac{1}{\\left(2 \\sin{\\left (x \\right )} - 1\\right)^{2} \\cos^{2}{\\left (x \\right )}}$$"
      ],
      "text/plain": [
       "           1           \n",
       "───────────────────────\n",
       "              2    2   \n",
       "(2⋅sin(x) - 1) ⋅cos (x)"
      ]
     },
     "execution_count": 19,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "f.simplify()"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 20,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "image/png": "iVBORw0KGgoAAAANSUhEUgAAA/0AAAAyBAMAAAD8R6avAAAAMFBMVEX///8AAAAAAAAAAAAAAAAA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\n",
      "text/latex": [
       "$$\\frac{\\sin{\\left (2 x \\right )}}{\\sin^{3}{\\left (2 x \\right )} - \\sin^{2}{\\left (2 x \\right )} \\cos{\\left (x \\right )} - \\sin{\\left (2 x \\right )} \\cos^{2}{\\left (x \\right )} + \\cos^{3}{\\left (x \\right )}} + \\frac{\\cos{\\left (x \\right )}}{\\sin^{3}{\\left (2 x \\right )} - \\sin^{2}{\\left (2 x \\right )} \\cos{\\left (x \\right )} - \\sin{\\left (2 x \\right )} \\cos^{2}{\\left (x \\right )} + \\cos^{3}{\\left (x \\right )}}$$"
      ],
      "text/plain": [
       "                         sin(2⋅x)                                             \n",
       "───────────────────────────────────────────────────────── + ──────────────────\n",
       "   3           2                           2         3         3           2  \n",
       "sin (2⋅x) - sin (2⋅x)⋅cos(x) - sin(2⋅x)⋅cos (x) + cos (x)   sin (2⋅x) - sin (2\n",
       "\n",
       "        cos(x)                         \n",
       "───────────────────────────────────────\n",
       "                         2         3   \n",
       "⋅x)⋅cos(x) - sin(2⋅x)⋅cos (x) + cos (x)"
      ]
     },
     "execution_count": 20,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "f.expand()"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 21,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "image/png": "iVBORw0KGgoAAAANSUhEUgAABLUAAAAyBAMAAABYEfR4AAAAMFBMVEX///8AAAAAAAAAAAAAAAAA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\nmecS4X5cozFNF4zp0xdil0BcMFqwOvz/E0c0S9+e4EQAAAAASUVORK5CYII=\n",
      "text/latex": [
       "$$\\frac{2 \\sin{\\left (x \\right )} \\cos{\\left (x \\right )}}{8 \\sin^{3}{\\left (x \\right )} \\cos^{3}{\\left (x \\right )} - 4 \\sin^{2}{\\left (x \\right )} \\cos^{3}{\\left (x \\right )} - 2 \\sin{\\left (x \\right )} \\cos^{3}{\\left (x \\right )} + \\cos^{3}{\\left (x \\right )}} + \\frac{\\cos{\\left (x \\right )}}{8 \\sin^{3}{\\left (x \\right )} \\cos^{3}{\\left (x \\right )} - 4 \\sin^{2}{\\left (x \\right )} \\cos^{3}{\\left (x \\right )} - 2 \\sin{\\left (x \\right )} \\cos^{3}{\\left (x \\right )} + \\cos^{3}{\\left (x \\right )}}$$"
      ],
      "text/plain": [
       "                         2⋅sin(x)⋅cos(x)                                      \n",
       "────────────────────────────────────────────────────────────────── + ─────────\n",
       "     3       3           2       3                  3         3           3   \n",
       "8⋅sin (x)⋅cos (x) - 4⋅sin (x)⋅cos (x) - 2⋅sin(x)⋅cos (x) + cos (x)   8⋅sin (x)\n",
       "\n",
       "                     cos(x)                              \n",
       "─────────────────────────────────────────────────────────\n",
       "    3           2       3                  3         3   \n",
       "⋅cos (x) - 4⋅sin (x)⋅cos (x) - 2⋅sin(x)⋅cos (x) + cos (x)"
      ]
     },
     "execution_count": 21,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "f.expand(trig=True)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 22,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "image/png": "iVBORw0KGgoAAAANSUhEUgAAAWoAAAA1BAMAAACaWbxGAAAAMFBMVEX///8AAAAAAAAAAAAAAAAA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\nYII=\n",
      "text/latex": [
       "$$\\frac{2 \\sin{\\left (x \\right )} \\cos{\\left (x \\right )} + \\cos{\\left (x \\right )}}{\\left(\\sin{\\left (2 x \\right )} - \\cos{\\left (x \\right )}\\right) \\left(\\sin^{2}{\\left (2 x \\right )} - \\cos^{2}{\\left (x \\right )}\\right)}$$"
      ],
      "text/plain": [
       "         2⋅sin(x)⋅cos(x) + cos(x)        \n",
       "─────────────────────────────────────────\n",
       "                    ⎛   2           2   ⎞\n",
       "(sin(2⋅x) - cos(x))⋅⎝sin (2⋅x) - cos (x)⎠"
      ]
     },
     "execution_count": 22,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "f.expand(trig=True, numer=True)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 23,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "image/png": "iVBORw0KGgoAAAANSUhEUgAAAlAAAAAyBAMAAABhb/gOAAAAMFBMVEX///8AAAAAAAAAAAAAAAAA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\noTE/x1GwZj1CZ234T4filNO2b3ZhAAAAAElFTkSuQmCC\n",
      "text/latex": [
       "$$\\frac{2 \\sin{\\left (x \\right )} \\cos{\\left (x \\right )} + \\cos{\\left (x \\right )}}{8 \\sin^{3}{\\left (x \\right )} \\cos^{3}{\\left (x \\right )} - 4 \\sin^{2}{\\left (x \\right )} \\cos^{3}{\\left (x \\right )} - 2 \\sin{\\left (x \\right )} \\cos^{3}{\\left (x \\right )} + \\cos^{3}{\\left (x \\right )}}$$"
      ],
      "text/plain": [
       "                     2⋅sin(x)⋅cos(x) + cos(x)                     \n",
       "──────────────────────────────────────────────────────────────────\n",
       "     3       3           2       3                  3         3   \n",
       "8⋅sin (x)⋅cos (x) - 4⋅sin (x)⋅cos (x) - 2⋅sin(x)⋅cos (x) + cos (x)"
      ]
     },
     "execution_count": 23,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "fe = f.expand(trig=True,numer=True).expand(trig=True,denom=True)\n",
    "fe"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 24,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "image/png": "iVBORw0KGgoAAAANSUhEUgAAAM0AAAAzBAMAAADGCpEzAAAAMFBMVEX///8AAAAAAAAAAAAAAAAA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\nwn/53fZdw8jzld/tPzHAXRIN8AjDAAAAAElFTkSuQmCC\n",
      "text/latex": [
       "$$\\frac{1}{\\left(2 \\sin{\\left (x \\right )} - 1\\right)^{2} \\cos^{2}{\\left (x \\right )}}$$"
      ],
      "text/plain": [
       "           1           \n",
       "───────────────────────\n",
       "              2    2   \n",
       "(2⋅sin(x) - 1) ⋅cos (x)"
      ]
     },
     "execution_count": 24,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "fe.factor()"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 25,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "image/png": "iVBORw0KGgoAAAANSUhEUgAAA/0AAAAyBAMAAAD8R6avAAAAMFBMVEX///8AAAAAAAAAAAAAAAAA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\n",
      "text/latex": [
       "$$\\frac{\\sin{\\left (2 x \\right )}}{\\sin^{3}{\\left (2 x \\right )} - \\sin^{2}{\\left (2 x \\right )} \\cos{\\left (x \\right )} - \\sin{\\left (2 x \\right )} \\cos^{2}{\\left (x \\right )} + \\cos^{3}{\\left (x \\right )}} + \\frac{\\cos{\\left (x \\right )}}{\\sin^{3}{\\left (2 x \\right )} - \\sin^{2}{\\left (2 x \\right )} \\cos{\\left (x \\right )} - \\sin{\\left (2 x \\right )} \\cos^{2}{\\left (x \\right )} + \\cos^{3}{\\left (x \\right )}}$$"
      ],
      "text/plain": [
       "                         sin(2⋅x)                                             \n",
       "───────────────────────────────────────────────────────── + ──────────────────\n",
       "   3           2                           2         3         3           2  \n",
       "sin (2⋅x) - sin (2⋅x)⋅cos(x) - sin(2⋅x)⋅cos (x) + cos (x)   sin (2⋅x) - sin (2\n",
       "\n",
       "        cos(x)                         \n",
       "───────────────────────────────────────\n",
       "                         2         3   \n",
       "⋅x)⋅cos(x) - sin(2⋅x)⋅cos (x) + cos (x)"
      ]
     },
     "execution_count": 25,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "f.expand(gibtsnich=True)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Auswertungen von Ausdrücken"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 26,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "image/png": "iVBORw0KGgoAAAANSUhEUgAAAIMAAAA0BAMAAABbfJaYAAAAMFBMVEX///8AAAAAAAAAAAAAAAAA\nAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAv3aB7AAAAD3RSTlMAEJmJqyLvZs27djJE\nVN3lLQH2AAAACXBIWXMAAA7EAAAOxAGVKw4bAAAC6UlEQVRIDe2Wz2sTQRTHv2t2k82vzSJFSi+N\n6cGTNCTiwYNWQfCki4IiSKkHfxyKBEH0GA8i+KtREBH8EateejHgH9CgR0GLnrUBEYpgTRDx0Mb4\nZmY3ZTcva6DXDGRm3vt+8ubHzuxbIKQU7tkh6iBSxNHLg3AhjOUkf4fIg0hW3VgfhAtl0q1QuZ+4\ntXiwK1nVbld2zGfJnX4PZ2nncNzxhILXcdtop5MPuBhTn4Gedf0xr+Nx5q6XXjekzZQRabr6W7zz\nk7rf7GONV5BYU1ry9ue7fmqwEFGaxTrGXpx6XqWVe/NxI+mX52v+oJxFqzD/aPnYd6vcK8ftgU7K\nDXxqR+xIK+30hiDPE9brd8YWXq9pMCtdr3ZnlcqKu4JJPnKXVh2dzmSmGnBK8xCwr84JQZ85BSy6\ng/q188Ck7XcxVuwMFp20/QoJhs0CK8x/Aq7EVe06Piw9xYmAIMyjoBH+Xwq5OkYWxnaUGNSYOMDM\njQGHruEOoLPZ0hxuom8HDO5OSiLmKJC790q5pJqTqgnUx4qPAZVdjUpAk6aRe1iDrga5xgJZLJcw\nKiXTnYyfGxH5RisLZ0/ilGSijUwDkaowLkpPsNoPnAX2CLfFjpGaxXgZ2owgvomqp1wAjtSwTfin\ne0TXQQvBLeobDZaYs0UIs07iPP1kCg2A2ldyiA+MRJYqjsBPGxEhPgC4FGrs3U7iFfrFaTYcgdQv\n2skyET8APoXurgNfCLCW+hDWFJBuEDFLY3gpdFRkz9Wb5KUSPU1JTbT1DcKXX3MkJptUUQgmhRoO\n4u2NEAyh9kiGoIUwKTTTkiHekyYWwhD4CBon1SCNtpNJodEKLPr09baTIZJZJBy1nfTcmBQaK2G5\nCkzQGOK5McR0sXAY2JIngubKpdA3uUckip0Vc2WIuU7nL2DWiOAPOAlUjIao+QMuFPds89dMAWHX\nTBHymuG+MrhaXfa4w2nSpy47+FeOJFR0/pUjATf6Zl98/wBDlfAwmEL4NgAAAABJRU5ErkJggg==\n",
      "text/latex": [
       "$$\\frac{9 x^{2} - 5}{\\left(x - 3\\right) \\left(x - 2\\right)}$$"
      ],
      "text/plain": [
       "       2       \n",
       "    9⋅x  - 5   \n",
       "───────────────\n",
       "(x - 3)⋅(x - 2)"
      ]
     },
     "execution_count": 26,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "x = Symbol('x')\n",
    "a = (9*x**2-5)/((x-2)*(x-3))\n",
    "a"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 27,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "image/png": "iVBORw0KGgoAAAANSUhEUgAAACAAAAAqBAMAAAAgz0ShAAAAMFBMVEX///8AAAAAAAAAAAAAAAAA\nAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAv3aB7AAAAD3RSTlMAVO8Qq5l2zWYiibvd\nRDIcHY1cAAAACXBIWXMAAA7EAAAOxAGVKw4bAAABCElEQVQoFWNgAINZqzYyMJiu8gRzhAwYOAMY\n5BuYNRgmTWBgYHb1N2Bg+8jAf4BFgYElAKQk34CBV5fBvoC/gIHrA1QASMk32G9gYPsHF2A+xcAE\nVPEXJsC5Jhiknv0HTICBYdEFhjIGqY8IASZlBo60VzAzOCcwcIMkWT5DVfB/hgiwO0AFmDYw8Pzk\n0GR4D3Qp2GEcDQzyCWzFzBUgfoh+SQPD1dC9DAxLQy+ABAYJ+I8GaOksztDdD1DMN4FGCVzQmYFB\nCc4BMXQYGPwfIIvUG6AJACXPGyCrYGDg/YXKZ+BxQBMIReOzBaAJiDIAIxwJ8AUwsKEIzFm11AdJ\nnoGh/v//PxABACNkTvWAxxf+AAAAAElFTkSuQmCC\n",
      "text/latex": [
       "$$\\frac{139}{2}$$"
      ],
      "text/plain": [
       "139/2"
      ]
     },
     "execution_count": 27,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "a.subs(x, 4)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 28,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "image/png": "iVBORw0KGgoAAAANSUhEUgAAAEoAAAA2BAMAAACSHhL4AAAAMFBMVEX///8AAAAAAAAAAAAAAAAA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\nTwAAAABJRU5ErkJggg==\n",
      "text/latex": [
       "$$\\frac{\\left(x - y\\right)^{3}}{x + y}$$"
      ],
      "text/plain": [
       "       3\n",
       "(x - y) \n",
       "────────\n",
       " x + y  "
      ]
     },
     "execution_count": 28,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "y = Symbol('y')\n",
    "b = (x-y)**3 / (x+y)\n",
    "b"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 29,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "image/png": "iVBORw0KGgoAAAANSUhEUgAAAAoAAAAOBAMAAADkjZCYAAAALVBMVEX///8AAAAAAAAAAAAAAAAA\nAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAADAOrOgAAAADnRSTlMAEJmJqyLvZs27djJEVPfV\nizYAAAAJcEhZcwAADsQAAA7EAZUrDhsAAABVSURBVAgdY2AQMnZlYGBMZwgLYGBJYGBRYOBrYGB+\nwCA3gYHrNQMbkP0GxOV4zsDQxXD5LQMD+6rNrxmAgOUhiORwYGBPZdgXwMDVyNgO5BspHWAAAM1O\nEeMqKO2rAAAAAElFTkSuQmCC\n",
      "text/latex": [
       "$$9$$"
      ],
      "text/plain": [
       "9"
      ]
     },
     "execution_count": 29,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "b.subs(x, 3).subs(y, 0)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 30,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "image/png": "iVBORw0KGgoAAAANSUhEUgAAAIMAAAA0BAMAAABbfJaYAAAAMFBMVEX///8AAAAAAAAAAAAAAAAA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\n",
      "text/latex": [
       "$$\\frac{9 x^{2} - 5}{\\left(x - 3\\right) \\left(x - 2\\right)}$$"
      ],
      "text/plain": [
       "       2       \n",
       "    9⋅x  - 5   \n",
       "───────────────\n",
       "(x - 3)⋅(x - 2)"
      ]
     },
     "execution_count": 30,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "a"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 31,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "image/png": "iVBORw0KGgoAAAANSUhEUgAAABMAAAAPBAMAAAD0aukfAAAAMFBMVEX///8AAAAAAAAAAAAAAAAA\nAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAv3aB7AAAAD3RSTlMAMs27ZhBURHar3e+J\nmSK3E9V5AAAACXBIWXMAAA7EAAAOxAGVKw4bAAAAfElEQVQIHWNgYGBgVHZNB1IgEBE4qwHMwEEE\ngMRBBOvs3w4MDIdvNjAwMAtwfGIISmCQYWBwZGDIN1BkYGB5wACU4PkFJBgcQEwGexDRAGZafwdq\nd2BwZWDgOLf7AQPHAwZeBoazDPLbGXyAEncnJzCw3t99AKgcbBGIAAAqoxtVgwlygwAAAABJRU5E\nrkJggg==\n",
      "text/latex": [
       "$$\\tilde{\\infty}$$"
      ],
      "text/plain": [
       "zoo"
      ]
     },
     "execution_count": 31,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "a.subs(x, 2)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 32,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "image/png": "iVBORw0KGgoAAAANSUhEUgAAAEsAAAA0BAMAAAAwFNjNAAAAMFBMVEX///8AAAAAAAAAAAAAAAAA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\nQ7U+z+OWxJs5fRWeAL8qnqf+gYdYAAAAAElFTkSuQmCC\n",
      "text/latex": [
       "$$\\frac{1}{\\cos{\\left (\\frac{\\pi x}{2} \\right )}}$$"
      ],
      "text/plain": [
       "   1    \n",
       "────────\n",
       "   ⎛π⋅x⎞\n",
       "cos⎜───⎟\n",
       "   ⎝ 2 ⎠"
      ]
     },
     "execution_count": 32,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "c = 1/cos(x*pi/2)\n",
    "c"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 33,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "image/png": "iVBORw0KGgoAAAANSUhEUgAAABMAAAAPBAMAAAD0aukfAAAAMFBMVEX///8AAAAAAAAAAAAAAAAA\nAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAv3aB7AAAAD3RSTlMAMs27ZhBURHar3e+J\nmSK3E9V5AAAACXBIWXMAAA7EAAAOxAGVKw4bAAAAfElEQVQIHWNgYGBgVHZNB1IgEBE4qwHMwEEE\ngMRBBOvs3w4MDIdvNjAwMAtwfGIISmCQYWBwZGDIN1BkYGB5wACU4PkFJBgcQEwGexDRAGZafwdq\nd2BwZWDgOLf7AQPHAwZeBoazDPLbGXyAEncnJzCw3t99AKgcbBGIAAAqoxtVgwlygwAAAABJRU5E\nrkJggg==\n",
      "text/latex": [
       "$$\\tilde{\\infty}$$"
      ],
      "text/plain": [
       "zoo"
      ]
     },
     "execution_count": 33,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "c.subs(x,1)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Grenzwerte"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 34,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "image/png": "iVBORw0KGgoAAAANSUhEUgAAAIMAAAA0BAMAAABbfJaYAAAAMFBMVEX///8AAAAAAAAAAAAAAAAA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\n",
      "text/latex": [
       "$$\\frac{9 x^{2} - 5}{\\left(x - 3\\right) \\left(x - 2\\right)}$$"
      ],
      "text/plain": [
       "       2       \n",
       "    9⋅x  - 5   \n",
       "───────────────\n",
       "(x - 3)⋅(x - 2)"
      ]
     },
     "execution_count": 34,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "a"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 35,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "image/png": "iVBORw0KGgoAAAANSUhEUgAAACMAAAALBAMAAAAHCCkxAAAAMFBMVEX///8AAAAAAAAAAAAAAAAA\nAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAv3aB7AAAAD3RSTlMAEM3dMqvvZom7mXZU\nIkRJD0iWAAAACXBIWXMAAA7EAAAOxAGVKw4bAAAAfklEQVQIHWNggAMBEAtMQIUYw74VMDB0Lt0A\nV8LA6cD9iUHoAIMHQqiEgeH8BBUGBvYLDELGIKDCANTA8RmkC6gdCkC8+SAChCEAxJr2j4GBsQAm\nwlDIwMDdm3aBgfsCXIiLgaGLwT+PoQIuwsC4KvIAA+P6tAaEENThAgwMAMSLGqu/gFQwAAAAAElF\nTkSuQmCC\n",
      "text/latex": [
       "$$-\\infty$$"
      ],
      "text/plain": [
       "-∞"
      ]
     },
     "execution_count": 35,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "a.limit(x, 2)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 36,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "image/png": "iVBORw0KGgoAAAANSUhEUgAAABMAAAALBAMAAABv+6sJAAAAMFBMVEX///8AAAAAAAAAAAAAAAAA\nAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAv3aB7AAAAD3RSTlMAEKvvZom7mXYyzVQi\n3UQ6SGZXAAAACXBIWXMAAA7EAAAOxAGVKw4bAAAAaklEQVQIHWNgYBBgAAIQwaj82YGBIayogYGB\nbQLHLwapDQxTGRg8GRj2J6xkYGA5wACUYP0LJBgcQEyGfBDRAGYm/wNqd2BwZGDgiDE+wMBxgIGd\ngSGcYb4dgytQolxtAwNjvXEAUDncNgBJUBUwaYAbUgAAAABJRU5ErkJggg==\n",
      "text/latex": [
       "$$\\infty$$"
      ],
      "text/plain": [
       "∞"
      ]
     },
     "execution_count": 36,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "a.limit(x, 2, dir='-')"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 37,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "image/png": "iVBORw0KGgoAAAANSUhEUgAAAEgAAAAtBAMAAAD/+jCdAAAAMFBMVEX///8AAAAAAAAAAAAAAAAA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\nmH8BdkWksadATsUAAAAASUVORK5CYII=\n",
      "text/latex": [
       "$$\\frac{\\cos{\\left (\\pi x \\right )}}{2 x - 1}$$"
      ],
      "text/plain": [
       "cos(π⋅x)\n",
       "────────\n",
       "2⋅x - 1 "
      ]
     },
     "execution_count": 37,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "a = cos(pi*x)/(2*x-1)\n",
    "a"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 38,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "image/png": "iVBORw0KGgoAAAANSUhEUgAAAB4AAAAlBAMAAACno0T/AAAAMFBMVEX///8AAAAAAAAAAAAAAAAA\nAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAv3aB7AAAAD3RSTlMAEM3dMnaZiVQiZrur\nRO9IrJcIAAAACXBIWXMAAA7EAAAOxAGVKw4bAAAAoElEQVQoFWNggAHW/0CQAOMxMJR7RC6dieCy\nNLBPCEFwgSxuhoMofB8GAxR+FsNHZD7jR4a/yHxWA4bnyHzuBQzHkfkDxBYyBgEVBn5QcPz/T0NX\ncKbtdkA2XoaBHSWAgKH7Dln+MQPDeWQN+gKofKBaewFkDQy831G4DMwHUPlpqFzWBFR+GwPnBCQR\nlgQGVmT+5PKyw0jSDPr//39mAABFUyhaLtLPEAAAAABJRU5ErkJggg==\n",
      "text/latex": [
       "$$- \\frac{\\pi}{2}$$"
      ],
      "text/plain": [
       "-π \n",
       "───\n",
       " 2 "
      ]
     },
     "execution_count": 38,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "a.limit(x, Rational(1,2))"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 39,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "image/png": "iVBORw0KGgoAAAANSUhEUgAAACkAAAAOBAMAAABA5yhLAAAAMFBMVEX///8AAAAAAAAAAAAAAAAA\nAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAv3aB7AAAAD3RSTlMAMqu73RDvmc1mVEQi\niXZo93taAAAACXBIWXMAAA7EAAAOxAGVKw4bAAAAt0lEQVQYGWMQsndlYDD7UsAAAgiO8CcGBtYG\nsCADA5wjqb+AgQGilIEBzpEMS0UShXEkuT8jicI4kgz1E0AmzHv7AGgClANksPwBinL5MBSBRMEc\nEIPxB0g0kWF/AIwDEmW4/wBoAuPb+gNwDojB8qeAgbX5gLwAjANWywP0GrMDg7xoAFAJiAMWZXhv\nwMB/gWG9tABQFMRhYFgLxBwGDGwGDPclA6AcBuH/XUBdBQw86m/j+mAcAHj0OnpfuCbeAAAAAElF\nTkSuQmCC\n",
      "text/latex": [
       "$$\\mathrm{NaN}$$"
      ],
      "text/plain": [
       "nan"
      ]
     },
     "execution_count": 39,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "a.subs(x, Rational(1,2))"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 40,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "image/png": "iVBORw0KGgoAAAANSUhEUgAAAEwAAAAwBAMAAABJWYGiAAAAMFBMVEX///8AAAAAAAAAAAAAAAAA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\nAABJRU5ErkJggg==\n",
      "text/latex": [
       "$$\\frac{n^{- n}}{\\sqrt{n}} e^{n} n!$$"
      ],
      "text/plain": [
       " -n  n   \n",
       "n  ⋅ℯ ⋅n!\n",
       "─────────\n",
       "    √n   "
      ]
     },
     "execution_count": 40,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "n = Symbol('n')\n",
    "b = factorial(n)*exp(n)/n**n/sqrt(n)\n",
    "b"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 41,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "image/png": "iVBORw0KGgoAAAANSUhEUgAAAJkAAAAyBAMAAAC9qWSkAAAAMFBMVEX///8AAAAAAAAAAAAAAAAA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\n",
      "text/latex": [
       "$$\\lim_{n \\to \\infty}\\left(\\frac{n^{- n}}{\\sqrt{n}} e^{n} n!\\right)$$"
      ],
      "text/plain": [
       "     -n  n   \n",
       "    n  ⋅ℯ ⋅n!\n",
       "lim ─────────\n",
       "n─→∞    √n   "
      ]
     },
     "execution_count": 41,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "L = Limit(b, n, oo)   # Träger Operator\n",
    "L"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 42,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "image/png": "iVBORw0KGgoAAAANSUhEUgAAADkAAAAYBAMAAACyxkglAAAAMFBMVEX///8AAAAAAAAAAAAAAAAA\nAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAv3aB7AAAAD3RSTlMAInarRM2ZVBDdiWbv\nuzJCz3LGAAAACXBIWXMAAA7EAAAOxAGVKw4bAAABJ0lEQVQoFWWQMUvDUBSFTxoaGw1WB0VwqoPg\n1P4DM7iL/yDg4CSIv+CtThVcxClOggXt7OTSvbsgTjqqS+mkntv7krz3vJDce8555Hs3gFb0K/Vl\nVdjiPHRcveGKf/OJ5yinBnUuvTTgJGdeGnAugM3T1/qEz8EInQEecxsHHMrYIHu2acBpjZHxwI9N\nyanqgEMKtOdNOqoytCYcdxYynasr2D7/6wzY6ueAbntYakrO8sP5dm+NcjjFitpXVJbzgdvWGyWS\nb6RmMRRwOFMuwYpmeJKOXT41Z3XAm0rtm2tpSwXWG04y7hpx8XlXSHsHeg1naLq5uGhPDN/Rzf0x\nr15x9pCVdPlJ7sUzXFAWs5wXxKXYwJE2fVuOazmzchzDG5XjWa5wOX8DxFCkyftiQAAAAABJRU5E\nrkJggg==\n",
      "text/latex": [
       "$$\\sqrt{2} \\sqrt{\\pi}$$"
      ],
      "text/plain": [
       "√2⋅√π"
      ]
     },
     "execution_count": 42,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "L.doit()"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Stirlingsche Formel"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Ableitungen"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 43,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "image/png": "iVBORw0KGgoAAAANSUhEUgAAABYAAAAQBAMAAADgw5IVAAAAMFBMVEX///8AAAAAAAAAAAAAAAAA\nAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAv3aB7AAAAD3RSTlMAEHarIkSJZt3NVLsy\nme8Q6PJIAAAACXBIWXMAAA7EAAAOxAGVKw4bAAAAdklEQVQIHWNgAAP264d0ISwGBvb4Aj4HKMdR\ng4FfACZhxBAHYzK8YbjLA+WwfGBoXQNlsxowqBUwMAgpu6opQIQYE9ibOCZA2GwCbB+5HKDiDJwb\nICwQya+AYJ8/AGNzCegzMEGtj1+wn8EdKiGsJHQdaAsIAAAS7RLOxnYqlAAAAABJRU5ErkJggg==\n",
      "text/latex": [
       "$$x^{n}$$"
      ],
      "text/plain": [
       " n\n",
       "x "
      ]
     },
     "execution_count": 43,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "f = x**n\n",
    "f"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 44,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "image/png": "iVBORw0KGgoAAAANSUhEUgAAACQAAAArBAMAAADieDd+AAAAMFBMVEX///8AAAAAAAAAAAAAAAAA\nAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAv3aB7AAAAD3RSTlMARImrInaZuxAyVN3N\nZu80gH0WAAAACXBIWXMAAA7EAAAOxAGVKw4bAAAA30lEQVQoFWNgQAK8uVPDkLggJq/8BT4HVDGP\naAb+BlQhBlMGOTQRhpcMaZyoYiwfGDauZGBgVHYJNWVgaA12DApgNGAIusDAwMZeytDvwKHAu5X7\nAFTPLCDLfwJTA9MPLgeoUEO/AMN9Bg4GngKoAJBaz8BgA6T4AxBCOQwMfzkYGOZPQAg9YuD6wcTV\nEM/A2gAVZPnIwHTgpvyCegZfmDLGDQwstguaQ1tzgS4a3OA/Bhjc7iXWdeA0gaIYJU1AZFDSBEQI\nNU1ADUBOE1Ah5DQBFkJJExBFKGkCIoScJgBRW0O7X7HIPwAAAABJRU5ErkJggg==\n",
      "text/latex": [
       "$$\\frac{n x^{n}}{x}$$"
      ],
      "text/plain": [
       "   n\n",
       "n⋅x \n",
       "────\n",
       " x  "
      ]
     },
     "execution_count": 44,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "f.diff(x)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 45,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "image/png": "iVBORw0KGgoAAAANSUhEUgAAADYAAAAUBAMAAAA0D9PTAAAAMFBMVEX///8AAAAAAAAAAAAAAAAA\nAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAv3aB7AAAAD3RSTlMARImrInaZuxAyVN3N\nZu80gH0WAAAACXBIWXMAAA7EAAAOxAGVKw4bAAAA1UlEQVQoFWNgwAXm4JJgYJhmjVuOYTXxcry5\nU8PgqtH08cpf4HOASaLJeUQz8DfgkGMwZZBj4CoHggoGDLe8ZEjjxKGP5QPDxpUwuXUwBoRmNGAI\nugAVcttvysCo7BJqysDQGuwYFICqkoGBjb2Uod+BQ4F3K/cBdLlZQCH/CUwNTD+4HNDlGvoFGO4z\ncDDwFKDLAPnrGRhsgBQ/hm1AwRwGhr8cDAzzJ2DR94iB6wcTV0M8A2sDuizLRwamAzflF9Qz+KJL\nMTBuYGCxXdAc2pp7AV0OACPFL1JXJc+pAAAAAElFTkSuQmCC\n",
      "text/latex": [
       "$$n x^{n - 1}$$"
      ],
      "text/plain": [
       "   n - 1\n",
       "n⋅x     "
      ]
     },
     "execution_count": 45,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "f.diff(x).powsimp()"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 46,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "image/png": "iVBORw0KGgoAAAANSUhEUgAAAKsAAAArBAMAAAD1dmXyAAAAMFBMVEX///8AAAAAAAAAAAAAAAAA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\n7no6qzQ3+3nXo9mB/wHfwcDAUEJHSwAAAABJRU5ErkJggg==\n",
      "text/latex": [
       "$$\\frac{n x^{n}}{x^{3}} \\left(n - 2\\right) \\left(n - 1\\right)$$"
      ],
      "text/plain": [
       "   n                \n",
       "n⋅x ⋅(n - 2)⋅(n - 1)\n",
       "────────────────────\n",
       "          3         \n",
       "         x          "
      ]
     },
     "execution_count": 46,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "f.diff(x,x,x).factor()"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 47,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "image/png": "iVBORw0KGgoAAAANSUhEUgAAAKsAAAArBAMAAAD1dmXyAAAAMFBMVEX///8AAAAAAAAAAAAAAAAA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\n7no6qzQ3+3nXo9mB/wHfwcDAUEJHSwAAAABJRU5ErkJggg==\n",
      "text/latex": [
       "$$\\frac{n x^{n}}{x^{3}} \\left(n - 2\\right) \\left(n - 1\\right)$$"
      ],
      "text/plain": [
       "   n                \n",
       "n⋅x ⋅(n - 2)⋅(n - 1)\n",
       "────────────────────\n",
       "          3         \n",
       "         x          "
      ]
     },
     "execution_count": 47,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "f.diff(x,3).factor()"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 48,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "ename": "ValueError",
     "evalue": "\nSince there is more than one variable in the expression, the\nvariable(s) of differentiation must be supplied to differentiate x**n",
     "output_type": "error",
     "traceback": [
      "\u001b[1;31m---------------------------------------------------------------------------\u001b[0m",
      "\u001b[1;31mValueError\u001b[0m                                Traceback (most recent call last)",
      "\u001b[1;32m<ipython-input-48-90cbeb354bfa>\u001b[0m in \u001b[0;36m<module>\u001b[1;34m()\u001b[0m\n\u001b[1;32m----> 1\u001b[1;33m \u001b[0mf\u001b[0m\u001b[1;33m.\u001b[0m\u001b[0mdiff\u001b[0m\u001b[1;33m(\u001b[0m\u001b[1;33m)\u001b[0m\u001b[1;33m\u001b[0m\u001b[0m\n\u001b[0m",
      "\u001b[1;32m/az076/anaconda3/lib/python3.5/site-packages/sympy/core/expr.py\u001b[0m in \u001b[0;36mdiff\u001b[1;34m(self, *symbols, **assumptions)\u001b[0m\n\u001b[0;32m   2864\u001b[0m         \u001b[0mnew_symbols\u001b[0m \u001b[1;33m=\u001b[0m \u001b[0mlist\u001b[0m\u001b[1;33m(\u001b[0m\u001b[0mmap\u001b[0m\u001b[1;33m(\u001b[0m\u001b[0msympify\u001b[0m\u001b[1;33m,\u001b[0m \u001b[0msymbols\u001b[0m\u001b[1;33m)\u001b[0m\u001b[1;33m)\u001b[0m  \u001b[1;31m# e.g. x, 2, y, z\u001b[0m\u001b[1;33m\u001b[0m\u001b[0m\n\u001b[0;32m   2865\u001b[0m         \u001b[0massumptions\u001b[0m\u001b[1;33m.\u001b[0m\u001b[0msetdefault\u001b[0m\u001b[1;33m(\u001b[0m\u001b[1;34m\"evaluate\"\u001b[0m\u001b[1;33m,\u001b[0m \u001b[1;32mTrue\u001b[0m\u001b[1;33m)\u001b[0m\u001b[1;33m\u001b[0m\u001b[0m\n\u001b[1;32m-> 2866\u001b[1;33m         \u001b[1;32mreturn\u001b[0m \u001b[0mDerivative\u001b[0m\u001b[1;33m(\u001b[0m\u001b[0mself\u001b[0m\u001b[1;33m,\u001b[0m \u001b[1;33m*\u001b[0m\u001b[0mnew_symbols\u001b[0m\u001b[1;33m,\u001b[0m \u001b[1;33m**\u001b[0m\u001b[0massumptions\u001b[0m\u001b[1;33m)\u001b[0m\u001b[1;33m\u001b[0m\u001b[0m\n\u001b[0m\u001b[0;32m   2867\u001b[0m \u001b[1;33m\u001b[0m\u001b[0m\n\u001b[0;32m   2868\u001b[0m     \u001b[1;31m###########################################################################\u001b[0m\u001b[1;33m\u001b[0m\u001b[1;33m\u001b[0m\u001b[0m\n",
      "\u001b[1;32m/az076/anaconda3/lib/python3.5/site-packages/sympy/core/function.py\u001b[0m in \u001b[0;36m__new__\u001b[1;34m(cls, expr, *variables, **assumptions)\u001b[0m\n\u001b[0;32m   1035\u001b[0m                         \u001b[0mSince\u001b[0m \u001b[0mthere\u001b[0m \u001b[1;32mis\u001b[0m \u001b[0mmore\u001b[0m \u001b[0mthan\u001b[0m \u001b[0mone\u001b[0m \u001b[0mvariable\u001b[0m \u001b[1;32min\u001b[0m \u001b[0mthe\u001b[0m\u001b[1;33m\u001b[0m\u001b[0m\n\u001b[0;32m   1036\u001b[0m                         \u001b[0mexpression\u001b[0m\u001b[1;33m,\u001b[0m \u001b[0mthe\u001b[0m \u001b[0mvariable\u001b[0m\u001b[1;33m(\u001b[0m\u001b[0ms\u001b[0m\u001b[1;33m)\u001b[0m \u001b[0mof\u001b[0m \u001b[0mdifferentiation\u001b[0m\u001b[1;33m\u001b[0m\u001b[0m\n\u001b[1;32m-> 1037\u001b[1;33m                         must be supplied to differentiate %s''' % expr))\n\u001b[0m\u001b[0;32m   1038\u001b[0m \u001b[1;33m\u001b[0m\u001b[0m\n\u001b[0;32m   1039\u001b[0m         \u001b[1;31m# Standardize the variables by sympifying them and making appending a\u001b[0m\u001b[1;33m\u001b[0m\u001b[1;33m\u001b[0m\u001b[0m\n",
      "\u001b[1;31mValueError\u001b[0m: \nSince there is more than one variable in the expression, the\nvariable(s) of differentiation must be supplied to differentiate x**n"
     ]
    }
   ],
   "source": [
    "f.diff()"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 49,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "image/png": "iVBORw0KGgoAAAANSUhEUgAAABYAAAAQBAMAAADgw5IVAAAAMFBMVEX///8AAAAAAAAAAAAAAAAA\nAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAv3aB7AAAAD3RSTlMAEHarIkSJZt3NVLsy\nme8Q6PJIAAAACXBIWXMAAA7EAAAOxAGVKw4bAAAAdklEQVQIHWNgAAP264d0ISwGBvb4Aj4HKMdR\ng4FfACZhxBAHYzK8YbjLA+WwfGBoXQNlsxowqBUwMAgpu6opQIQYE9ibOCZA2GwCbB+5HKDiDJwb\nICwQya+AYJ8/AGNzCegzMEGtj1+wn8EdKiGsJHQdaAsIAAAS7RLOxnYqlAAAAABJRU5ErkJggg==\n",
      "text/latex": [
       "$$x^{n}$$"
      ],
      "text/plain": [
       " n\n",
       "x "
      ]
     },
     "execution_count": 49,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "f.diff(x,0)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 50,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "image/png": "iVBORw0KGgoAAAANSUhEUgAAAOwAAAAZBAMAAADAot+NAAAAMFBMVEX///8AAAAAAAAAAAAAAAAA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\n/lPzB8tQEaizdO52AAAAAElFTkSuQmCC\n",
      "text/latex": [
       "$$- 2 x^{2} \\sin{\\left (2 x \\right )} + 2 x \\cos{\\left (2 x \\right )}$$"
      ],
      "text/plain": [
       "     2                        \n",
       "- 2⋅x ⋅sin(2⋅x) + 2⋅x⋅cos(2⋅x)"
      ]
     },
     "execution_count": 50,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "g = x**2*cos(2*x)\n",
    "g.diff()"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "So ist das Leben."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 51,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "image/png": "iVBORw0KGgoAAAANSUhEUgAAAEgAAAAtBAMAAAD/+jCdAAAAMFBMVEX///8AAAAAAAAAAAAAAAAA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\nmH8BdkWksadATsUAAAAASUVORK5CYII=\n",
      "text/latex": [
       "$$\\frac{\\cos{\\left (\\pi x \\right )}}{2 x - 1}$$"
      ],
      "text/plain": [
       "cos(π⋅x)\n",
       "────────\n",
       "2⋅x - 1 "
      ]
     },
     "execution_count": 51,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "a"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Wir berechnen den Grenzwert zu Fuß mit der Regel von de l'Hôpital"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 52,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "image/png": "iVBORw0KGgoAAAANSUhEUgAAAKsAAAAUBAMAAAAAWtCKAAAAMFBMVEX///8AAAAAAAAAAAAAAAAA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\n6/tyMWa5xuXLNp+MFlal2ens//7p/AHGwZdonDQ0YgAAAABJRU5ErkJggg==\n",
      "text/latex": [
       "$$\\left ( \\cos{\\left (\\pi x \\right )}, \\quad 2 x - 1\\right )$$"
      ],
      "text/plain": [
       "(cos(π⋅x), 2⋅x - 1)"
      ]
     },
     "execution_count": 52,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "zaehler = numer(a)    # numer ist keine Methode\n",
    "nenner = denom(a)\n",
    "zaehler, nenner"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 53,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "image/png": "iVBORw0KGgoAAAANSUhEUgAAAJkAAAAUBAMAAABsXIKPAAAAMFBMVEX///8AAAAAAAAAAAAAAAAA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\n4zeIKQAAAABJRU5ErkJggg==\n",
      "text/latex": [
       "$$\\left ( - \\pi \\sin{\\left (\\pi x \\right )}, \\quad 2\\right )$$"
      ],
      "text/plain": [
       "(-π⋅sin(π⋅x), 2)"
      ]
     },
     "execution_count": 53,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "d_zaehler = zaehler.diff(x)\n",
    "d_nenner = nenner.diff(x)\n",
    "d_zaehler, d_nenner"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 54,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "image/png": "iVBORw0KGgoAAAANSUhEUgAAAB4AAAAlBAMAAACno0T/AAAAMFBMVEX///8AAAAAAAAAAAAAAAAA\nAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAv3aB7AAAAD3RSTlMAEM3dMnaZiVQiZrur\nRO9IrJcIAAAACXBIWXMAAA7EAAAOxAGVKw4bAAAAoElEQVQoFWNggAHW/0CQAOMxMJR7RC6dieCy\nNLBPCEFwgSxuhoMofB8GAxR+FsNHZD7jR4a/yHxWA4bnyHzuBQzHkfkDxBYyBgEVBn5QcPz/T0NX\ncKbtdkA2XoaBHSWAgKH7Dln+MQPDeWQN+gKofKBaewFkDQy831G4DMwHUPlpqFzWBFR+GwPnBCQR\nlgQGVmT+5PKyw0jSDPr//39mAABFUyhaLtLPEAAAAABJRU5ErkJggg==\n",
      "text/latex": [
       "$$- \\frac{\\pi}{2}$$"
      ],
      "text/plain": [
       "-π \n",
       "───\n",
       " 2 "
      ]
     },
     "execution_count": 54,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "d_zaehler.subs(x, Rational(1,2)) / d_nenner.subs(x, Rational(1,2))"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## unbestimmte Integrale"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 55,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "image/png": "iVBORw0KGgoAAAANSUhEUgAAABYAAAAQBAMAAADgw5IVAAAAMFBMVEX///8AAAAAAAAAAAAAAAAA\nAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAv3aB7AAAAD3RSTlMAEHarIkSJZt3NVLsy\nme8Q6PJIAAAACXBIWXMAAA7EAAAOxAGVKw4bAAAAdklEQVQIHWNgAAP264d0ISwGBvb4Aj4HKMdR\ng4FfACZhxBAHYzK8YbjLA+WwfGBoXQNlsxowqBUwMAgpu6opQIQYE9ibOCZA2GwCbB+5HKDiDJwb\nICwQya+AYJ8/AGNzCegzMEGtj1+wn8EdKiGsJHQdaAsIAAAS7RLOxnYqlAAAAABJRU5ErkJggg==\n",
      "text/latex": [
       "$$x^{n}$$"
      ],
      "text/plain": [
       " n\n",
       "x "
      ]
     },
     "execution_count": 55,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "f"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 56,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "image/png": "iVBORw0KGgoAAAANSUhEUgAAALwAAAA/BAMAAACoQ7/uAAAAMFBMVEX///8AAAAAAAAAAAAAAAAA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\n+m3GDQkaj4g4+JxLmI4S2Hw4MJIH9D9OGIfJjTNLogAAAABJRU5ErkJggg==\n",
      "text/latex": [
       "$$\\begin{cases} \\log{\\left (x \\right )} & \\text{for}\\: n = -1 \\\\\\frac{x^{n + 1}}{n + 1} & \\text{otherwise} \\end{cases}$$"
      ],
      "text/plain": [
       "⎧log(x)  for n = -1\n",
       "⎪                  \n",
       "⎪ n + 1            \n",
       "⎨x                 \n",
       "⎪──────  otherwise \n",
       "⎪n + 1             \n",
       "⎩                  "
      ]
     },
     "execution_count": 56,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "f.integrate(x)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 57,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "image/png": "iVBORw0KGgoAAAANSUhEUgAAADEAAAAxBAMAAACBslwvAAAAMFBMVEX///8AAAAAAAAAAAAAAAAA\nAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAv3aB7AAAAD3RSTlMAEHarIkSJZt3NVLsy\nme8Q6PJIAAAACXBIWXMAAA7EAAAOxAGVKw4bAAABMklEQVQ4EWNgwAJOYhEDC521wCZjABJcgU1m\nA4YM+/VDuiBBLDLxBXwOWGUcNRj4BbDKMBgxxDEwz5z5cuYkdBe8YbjLg1UPyweG1jUwmXVABggI\nKbuqKbAaMKgVADkgt/n2G4HEGRgT2Js4JoCZQAIkAwNsAmwfuRxgPGQZRgZOJC5cCVgpvwJMBzp9\n/gC6CITPJaDPwCSATS5+wX4Gd2wSDMJKQtcLsMoMM8H/OMCHIedProfoTl4EERC+/BGHDAMTARnW\nZBclSDJmYICaBtXDy72JQd4BYi6qzClgavc/gE1GQD6AoR4kkTlzZt/MmZNBTKgL1jMwWIK4QIBq\nGsNNBobvjFhlnjFwfWTDJsPylYFtQg02GdYGBhbzBSgyvHY/zSECcBLmArgAnCECZwEZAMoIZKmV\nybVRAAAAAElFTkSuQmCC\n",
      "text/latex": [
       "$$\\frac{x^{n + 1}}{n + 1}$$"
      ],
      "text/plain": [
       " n + 1\n",
       "x     \n",
       "──────\n",
       "n + 1 "
      ]
     },
     "execution_count": 57,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "f.integrate(x, conds='none')"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 58,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "image/png": "iVBORw0KGgoAAAANSUhEUgAAAJMAAAAZBAMAAADH4AGYAAAAMFBMVEX///8AAAAAAAAAAAAAAAAA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\nHlFg6zkAAAAASUVORK5CYII=\n",
      "text/latex": [
       "$$\\sin{\\left (2 x \\right )} \\cos^{2}{\\left (3 x \\right )}$$"
      ],
      "text/plain": [
       "            2     \n",
       "sin(2⋅x)⋅cos (3⋅x)"
      ]
     },
     "execution_count": 58,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "f = sin(2*x)*cos(3*x)**2\n",
    "f"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 59,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "image/png": "iVBORw0KGgoAAAANSUhEUgAAAoYAAAArBAMAAAAXuVRTAAAAMFBMVEX///8AAAAAAAAAAAAAAAAA\nAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAv3aB7AAAAD3RSTlMAIom7VJlmdt1E7xDN\nqzIhoty3AAAACXBIWXMAAA7EAAAOxAGVKw4bAAAIxElEQVRoBe1aXYwb1RX+xnZ2bY/X6xCBQiqB\n42oJUGgWUYgKlBhBaVUe1iQsFUJUphKkqiphoqZ5ABoHqKASKBZqRcVDs6iEloSWURBKFCXEtFXV\nVvwsSCltqYgfiqARdDe/JV0Sc86dvzuzdzJj78wuD1zJnnvPPef7vnt27vycNQBolS9W6bCwTV9z\naSl5BR8cqFTKSdCcjcyJJHB7wtxT157uKaAv53u63e5kX5EhQZuq+DjEJfnpfwIHkme5GRhKhOWp\nlv5JIsC9gJKClb349+dbBr7cX2Ro1MLvZf0osKkTKnTuDrnG3DGUCLvLSvN8Gp+n87A1D4TphDjO\n+omREHJ0WNrIb9Sju/fteUHfkSGBqedCHJKfLjQyK6vJ0+jTiXFsX/gTcd2PV3YU6xu5IlZpg20F\nR0RTau3tgZ6LgdXVwNn5mzioyFaqNNSOU0F2tH+05fhXYHDX+Gzk8DWFwmypeFhh7tu0f7LvUGzG\nVCko+lfAlsDJoKC47d9XJyvb0k7GSTVV6x/tpzi/FRR9NzL/C5qbN/v9xt66kiwzozT3aRybQw6B\nMSOINje+Nii/58gxd8oDp6/VnS4wUJIGUjccJzV+m+QvdbMNHsTEgmVVCdrbjUDxiIgIdyz+2oXW\nRqGNX96xDENOgg6JdZnmJcBZla/YMW/aHc+RcdzikIvjcZIHkogRYY/KolIrA5+hTxRnr73WdlAt\nJFWLqKV4xIYBCiXsQ8q+IOlte2aqbPeAV6DVMFa3DHvdCalHOFJxyMWRXLxdV8SAqTsii1KtFzpw\n9Ar0SfddSbWQu8zYKFpcFgp6B9huG1bYHelIaxw8huGmZUpJZ6jrRThycUiF4zp7e0/gl2SIyhKm\n1ovtGRHFkIH8hGVULCRXznVoNpIWC4UOm4HfAAc5ktse8+D5zpaQew7725ZRn/DMWgPCkYtDKhxV\nGNmKf7zhajpEZQlTG8DCZqLINzF02nJRLOTDG643TMewFWv/tVDoQt6kl1jDzWGhZU2d13F8loqe\ns5fxF2fG7TAOpEKvg+O6+HqOiHy3O01zUVmUan3YAUOiKMy4OVQs5IVul2NDtBR/uK695BQ+3Pb+\nA7wnB2uCb4uhb3zzSdqA4pr6/o33a6sni3/9zsa3eXYDf+kPArseOHc9BY3w+KXKLdBXrG+BnGlo\n4uwuQ8ZhP2XziRA+0VngVatkEEafRnMhyM7IC9HXjFSxc+NFYE02lFKLPQm8CDRBrwsPNTR+O0jX\n6Qu5I9iJ7zUeoytBm9I1iiwKk8iuwvklmr2SPtqFt0MvDzySpfkf8PgZHCp9UMIq4WzhcHFIwiG3\ngOYTIbwis/jUBlCQ2a9RLATYWpYXsqxafLq4GemG0GSBqbRIPE8ZaOBrwFXACTJnqzyXncSXsNK4\njnZjkzK6A4NITyI9geE6gIfpA1zcShmpmQwl9Zs0KrQx1qFTc+o/7GzjUHFIwuEodfOJEE7RWbxq\n1Qxs9Wu0FvI45IW8iszpbBnF00KTBTZbC/2TxmrTtNij94JzeA3Aby75FoeNAwb+wL0iuWDLqpLI\nYRvDDRqaVbP8Dh2FUfaZos/+GvkeA4bL7GzjYLsh4+h/fpTaYx2aH7YliAuOTwTN98LiUyuxwGWZ\nnq3RXMiiGqSF6MeJeqoKHBGaWAg11YrNGfGtX9KtcQ4vl3JoXszEy6DI4c43joscjjo51CipnLEG\nY3AOt1IvQ0+Vw6PsbOZwsVkccnHYWdl8IoQP6Y7EAr9aJQMb/RrN1GzgKWchOVY71gKOCk08R02l\nxZwR31/AwEkph2Ivf53VF8UtP9ekpRi4p8V72cohndnDMyKHUx3G+AV9nPPwZnY297JZHHJx2Ffd\nfCKEU1QW+NSqGdjq1yj2cqYG+ls7C3HOw+NCkwWm0iLxfAP4m5TDdJ32JD1Dl24anMZ95j0l08BQ\nXcohXWHzdJs5nDE2YdAw7yl0rUF1G/2tf8bO5j1FFIckHAQ2nwjhF5XFrzaQhK+HHo3invIecK+8\nkFdpP+VryJ0UmiwwlRaJ51sGLuONbO1lfpZZWhn5e+5YeppftBeVCXMH0iU6Ce3zkJ5lBuoYa2yt\nrsY5BLWGPtqzSJUOdfB74Ww+E3FxSMYht4DmEyG8orL41QZQkNmvkR/KtAcrdzTlhSyrY3nxdyiU\nhSYLTKVF4nli3XdbH3Xf+aj77vX/n6R7cJOesbvdj/W379twN7kVOmS74EcXFQ+eevngqX+v/i3Z\neO/+fPyr2Ldx16116r9OH+z79nJoay5tsTMNGYeLQzIOu6mbT4RwisriV6tmEFafRl5Ige46TXkh\n+oH19KZf+QdYk42l1MKTlfUd20c60jua3JbKA6ufLXmMWtMztAZhOKKoI1WIZmHEwmLWl6TajIcm\nEgVHqBzpPMXe1qK2B9Ec3OW1rfAOxYjewOWmeFWn6TAcUdSRai4youjHwiLqS3JtxkMTiYIjZjvq\nB/hfQVfw9W12S5dkm96WR3b/MrsjjqqSEd1UQnBEUUeuuXgweRAHi6gvDUm1GS9LFAoRoXDcZCB3\n2AtnjzRPYr2ZsH2W2B1x9OLbU2E4oqgj11zsQOcYB4uotuSl2oyDLjpRKIIcKYf04qZufKd12p1O\nT+6EV8bJOxSHfvEj11xkAtGPh4VqxZ7ajIcmEgVHKBwph8Ovr7jNgzfvAyrqUNtiJEvM9SW6KczE\nzkI53P8n5KuxA/cAaP7ihypEiTauL1HbWo6dhXM4g0U7YgfuBVD84odqLgm3i1tE8Hj8JLyXm/yi\nvKBtO23j8cQV5OlModpM7I1ySMXTgROxA0cHXCyKOmbNJXpUr55W5UfUZnqNDfGnHA5NLOx52BW/\n+BE1lxCxc5k260tmbWYuOIpYymGRrocTiqn5Momijqi5JMko6kt4j2ozsbNQDvEW9lRjB44OKH7x\nwzWX6CF9eA5wfUnUZvoIPmPIHS+8Vkdq/N0zOiU8KX7xwzWXZHm4viRqM8nSfI7eewY+Bd/yLW8h\nwFKZAAAAAElFTkSuQmCC\n",
      "text/latex": [
       "$$\\frac{3}{16} \\sin{\\left (2 x \\right )} \\sin{\\left (3 x \\right )} \\cos{\\left (3 x \\right )} - \\frac{9}{32} \\sin^{2}{\\left (3 x \\right )} \\cos{\\left (2 x \\right )} - \\frac{7}{32} \\cos{\\left (2 x \\right )} \\cos^{2}{\\left (3 x \\right )}$$"
      ],
      "text/plain": [
       "                                    2                               2     \n",
       "3⋅sin(2⋅x)⋅sin(3⋅x)⋅cos(3⋅x)   9⋅sin (3⋅x)⋅cos(2⋅x)   7⋅cos(2⋅x)⋅cos (3⋅x)\n",
       "──────────────────────────── - ──────────────────── - ────────────────────\n",
       "             16                         32                     32         "
      ]
     },
     "execution_count": 59,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "I1 = f.integrate()   # warum darf diese Variable nicht I heißen?\n",
    "I1"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 60,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "image/png": "iVBORw0KGgoAAAANSUhEUgAAAJMAAAAZBAMAAADH4AGYAAAAMFBMVEX///8AAAAAAAAAAAAAAAAA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\nHlFg6zkAAAAASUVORK5CYII=\n",
      "text/latex": [
       "$$\\sin{\\left (2 x \\right )} \\cos^{2}{\\left (3 x \\right )}$$"
      ],
      "text/plain": [
       "            2     \n",
       "sin(2⋅x)⋅cos (3⋅x)"
      ]
     },
     "execution_count": 60,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "I1.diff(x)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 61,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "True"
      ]
     },
     "execution_count": 61,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "I1.diff(x) == f"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 62,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "image/png": "iVBORw0KGgoAAAANSUhEUgAAA00AAAAwBAMAAAA/eYydAAAAMFBMVEX///8AAAAAAAAAAAAAAAAA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\n9hEfnm/zcG/vGlEbP0LEG/m/2fJW4H/4mRnjtzt+1wAAAABJRU5ErkJggg==\n",
      "text/latex": [
       "$$- \\frac{\\sqrt{2}}{8} \\log{\\left (x^{2} - \\sqrt{2} x + 1 \\right )} + \\frac{\\sqrt{2}}{8} \\log{\\left (x^{2} + \\sqrt{2} x + 1 \\right )} + \\frac{\\sqrt{2}}{4} \\operatorname{atan}{\\left (\\sqrt{2} x - 1 \\right )} + \\frac{\\sqrt{2}}{4} \\operatorname{atan}{\\left (\\sqrt{2} x + 1 \\right )}$$"
      ],
      "text/plain": [
       "        ⎛ 2           ⎞         ⎛ 2           ⎞                               \n",
       "  √2⋅log⎝x  - √2⋅x + 1⎠   √2⋅log⎝x  + √2⋅x + 1⎠   √2⋅atan(√2⋅x - 1)   √2⋅atan(\n",
       "- ───────────────────── + ───────────────────── + ───────────────── + ────────\n",
       "            8                       8                     4                   \n",
       "\n",
       "         \n",
       "√2⋅x + 1)\n",
       "─────────\n",
       "4        "
      ]
     },
     "execution_count": 62,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "g = 1/(1+x**4)\n",
    "I2 = g.integrate(x)\n",
    "I2"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 63,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "image/png": "iVBORw0KGgoAAAANSUhEUgAAAr0AAAA/BAMAAAAf7W0RAAAAMFBMVEX///8AAAAAAAAAAAAAAAAA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\n1aPMxBisAAAAAElFTkSuQmCC\n",
      "text/latex": [
       "$$- \\frac{\\sqrt{2} \\left(2 x - \\sqrt{2}\\right)}{8 x^{2} - 8 \\sqrt{2} x + 8} + \\frac{\\sqrt{2} \\left(2 x + \\sqrt{2}\\right)}{8 x^{2} + 8 \\sqrt{2} x + 8} + \\frac{1}{2 \\left(\\sqrt{2} x + 1\\right)^{2} + 2} + \\frac{1}{2 \\left(\\sqrt{2} x - 1\\right)^{2} + 2}$$"
      ],
      "text/plain": [
       "    √2⋅(2⋅x - √2)       √2⋅(2⋅x + √2)              1                     1    \n",
       "- ───────────────── + ───────────────── + ─────────────────── + ──────────────\n",
       "    ⎛ 2           ⎞     ⎛ 2           ⎞     ⎛          2    ⎞     ⎛          2\n",
       "  8⋅⎝x  - √2⋅x + 1⎠   8⋅⎝x  + √2⋅x + 1⎠   2⋅⎝(√2⋅x + 1)  + 1⎠   2⋅⎝(√2⋅x - 1) \n",
       "\n",
       "     \n",
       "─────\n",
       "    ⎞\n",
       " + 1⎠"
      ]
     },
     "execution_count": 63,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "I2.diff(x)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 64,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "False"
      ]
     },
     "execution_count": 64,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "I2.diff() == g"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 65,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "True"
      ]
     },
     "execution_count": 65,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "I2.diff().ratsimp() ==g"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## bestimmte Integrale"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 66,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "image/png": "iVBORw0KGgoAAAANSUhEUgAAAJMAAAAZBAMAAADH4AGYAAAAMFBMVEX///8AAAAAAAAAAAAAAAAA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\nHlFg6zkAAAAASUVORK5CYII=\n",
      "text/latex": [
       "$$\\sin{\\left (2 x \\right )} \\cos^{2}{\\left (3 x \\right )}$$"
      ],
      "text/plain": [
       "            2     \n",
       "sin(2⋅x)⋅cos (3⋅x)"
      ]
     },
     "execution_count": 66,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "f"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 67,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "image/png": "iVBORw0KGgoAAAANSUhEUgAAAAsAAAAqBAMAAACXcryGAAAAMFBMVEX///8AAAAAAAAAAAAAAAAA\nAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAv3aB7AAAAD3RSTlMAVO8Qq5l2zWaJRCK7\n3TJS0mj0AAAACXBIWXMAAA7EAAAOxAGVKw4bAAAAeklEQVQYGWNggAEhAyCL2dUfRDEw5FODCtEv\naQCbRhbxHwQ+EKXVLDMKpC6AgUWAgYGxgYG7gIGB8wAD8wUGBq4/BpwNQDn/HzNASjj/F4MokbWf\nFjAwcDgw5CszMDBtYOD7yMBgDxT3APMYTgP1FTNwJABFnkQCjQYAox8hDpDfVpUAAAAASUVORK5C\nYII=\n",
      "text/latex": [
       "$$\\frac{1}{8}$$"
      ],
      "text/plain": [
       "1/8"
      ]
     },
     "execution_count": 67,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "I3 = f.integrate((x,0,pi/4))\n",
    "I3"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 68,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "image/png": "iVBORw0KGgoAAAANSUhEUgAAAMYAAABNBAMAAAAPwgZ1AAAAMFBMVEX///8AAAAAAAAAAAAAAAAA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\nwPS0KXel8B/O9y9bjocqpAAAAABJRU5ErkJggg==\n",
      "text/latex": [
       "$$\\int_{0}^{\\frac{\\pi}{4}} \\sin{\\left (2 x \\right )} \\cos^{2}{\\left (3 x \\right )}\\, dx$$"
      ],
      "text/plain": [
       "π                      \n",
       "─                      \n",
       "4                      \n",
       "⌠                      \n",
       "⎮             2        \n",
       "⎮ sin(2⋅x)⋅cos (3⋅x) dx\n",
       "⌡                      \n",
       "0                      "
      ]
     },
     "execution_count": 68,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "I4 = Integral(f, (x,0,pi/4))\n",
    "I4"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 69,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "image/png": "iVBORw0KGgoAAAANSUhEUgAAAAsAAAAqBAMAAACXcryGAAAAMFBMVEX///8AAAAAAAAAAAAAAAAA\nAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAv3aB7AAAAD3RSTlMAVO8Qq5l2zWaJRCK7\n3TJS0mj0AAAACXBIWXMAAA7EAAAOxAGVKw4bAAAAeklEQVQYGWNggAEhAyCL2dUfRDEw5FODCtEv\naQCbRhbxHwQ+EKXVLDMKpC6AgUWAgYGxgYG7gIGB8wAD8wUGBq4/BpwNQDn/HzNASjj/F4MokbWf\nFjAwcDgw5CszMDBtYOD7yMBgDxT3APMYTgP1FTNwJABFnkQCjQYAox8hDpDfVpUAAAAASUVORK5C\nYII=\n",
      "text/latex": [
       "$$\\frac{1}{8}$$"
      ],
      "text/plain": [
       "1/8"
      ]
     },
     "execution_count": 69,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "I4.doit()"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 70,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "image/png": "iVBORw0KGgoAAAANSUhEUgAAAC4AAAAPBAMAAABpZ+CXAAAAMFBMVEX///8AAAAAAAAAAAAAAAAA\nAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAv3aB7AAAAD3RSTlMAEJmJZjLNVN0i77ur\nRHZ72Yd1AAAACXBIWXMAAA7EAAAOxAGVKw4bAAAA+klEQVQYGW2PsUoDQRCGv9vEXJK74JE0YuNa\nWJkioHZCrC1CyBMcWNiIWFsdWNhYSDqryxtEC/tDtNW8QSCdjeEQlRTmnIl4oPg3u/vtzPz/QH17\nh1z9zVs4DnoncMjK5YKfBjgJ7RHD7BWKAwqhcOfuKcAPqA44eJRCL8FNhcNUYIT/RqKv5YTah16U\nl9Kcr1lq8x8uZznlvrMBcYslsRFJvWjc4oZ4RGx/8wv99CL+zDFWuZmpr5v7wgT8kMIn3jXFPCeu\n5aoaYlLdy4Tau/DtwbORUELOWe0679+8crbVjCqWfVm48fIAkqI5PNorZVkW0d9d1/5/9QV0nkXQ\nZV0n0AAAAABJRU5ErkJggg==\n",
      "text/latex": [
       "$$0.125$$"
      ],
      "text/plain": [
       "0.125000000000000"
      ]
     },
     "execution_count": 70,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "I4.n()  # numerische Integration"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 71,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "image/png": "iVBORw0KGgoAAAANSUhEUgAAAHoAAABKBAMAAACGDV16AAAAMFBMVEX///8AAAAAAAAAAAAAAAAA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\nfsr3xH8D0h7B4ctYZHwAAAAASUVORK5CYII=\n",
      "text/latex": [
       "$$\\int_{0}^{1} \\sqrt{e^{x} + 4}\\, dx$$"
      ],
      "text/plain": [
       "1               \n",
       "⌠               \n",
       "⎮    ________   \n",
       "⎮   ╱  x        \n",
       "⎮ ╲╱  ℯ  + 4  dx\n",
       "⌡               \n",
       "0               "
      ]
     },
     "execution_count": 71,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "I5 = Integral(sqrt(exp(x)+4), (x,0,1))\n",
    "I5"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 72,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "image/png": "iVBORw0KGgoAAAANSUhEUgAAAHoAAABKBAMAAACGDV16AAAAMFBMVEX///8AAAAAAAAAAAAAAAAA\nAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAv3aB7AAAAD3RSTlMAMs2Zq91U7yJ2iRBE\nu2Z2muS3AAAACXBIWXMAAA7EAAAOxAGVKw4bAAADBUlEQVRYCe1Vv2vUUBz/pPczl/Zy4B/Q43Rw\nUSqoQ12KdLCD9hDBQaU3dFAcWjqIi/UoTgr25LqUVhuFCmfRBqqoXS6LiCD0cOnYW1wq6NWW/rDV\n+pK8d3n2EvuagC59cPl+vp8fycvLSw6whpS2q6+jdDzrK0dD0f20j+X7r6vW72PCLBI/sVJgeL+K\nr0DomS5ubnCOvO5o4IQJZSsS4JHJNeELuRgjFRdSmBoOMG9gwhC+kItxQXchhak1YaeLUVpxIYUp\neYO3xrf/MngjxS1bPBk2+G53rG7yniG+EcCtWd50km8E8LcOzqT8cSpO8ILd/FYL5bxswAFbkubb\nOE+PwTWvKC4dG0lxtAWvUELNOYqyrTsNXthY1k9HDBteY6p8i6I5LhDltxq7bQlHWaievn+EUpNM\nIjXxk2vkKmvWGainNZY+yCRSm37Q5iOpEfJTSsU8pJpEaZZO5q10vP3BKpTRMx9Klq6u2jZ5gNRR\n8nuXj2vDiWzCpsHSb2Gl3+dbNjGLz4Xzlj5M53jPfM2zQPwGZlIvx0tTO9KalVYGEUpjCtP6VUvv\nq1FbuYKkBoQHR79QBniSyVzMZA6RPpqy0qFNRDTo9jwIPZ0mB3OE1hEhz0LtsLr6gc58FlYiVkHZ\nINoG1RfaKIgvY5FAVaM9LTTd29m51gWUNfQRQVqi6u0cBejWLxEY04ACo0hlqwaYu6Wcw6do/jr5\niI+ZHm6rtT41pyGnMZM3FTqctLkxmgrJC4nm1ZZavGrqzc5Waxogtw0UH1XNwkY9/Xz7HLlY1+M7\n40rXWPGrpcvOVy26zBJ8rad5kuHwL4aAww500BsHNqIIW71GSYBRawImT0trxVMSECY0AZOnpdvw\nlASEHgGPt+W7t7S7IrEvy+5WF0cy0OMO97ucUpja+TYLBy3jXGpvft49ibN8u0c8L93cY4K3330Y\nYOL8if49lktagIsuojdA+jJU3XdcWUIs5zstkb+uqv80+Ucs+E+vI+Y/Te5bNXxfG4HWHMVAzzvR\nfsr3xH8D0h7B4ctYZHwAAAAASUVORK5CYII=\n",
      "text/latex": [
       "$$\\int_{0}^{1} \\sqrt{e^{x} + 4}\\, dx$$"
      ],
      "text/plain": [
       "1               \n",
       "⌠               \n",
       "⎮    ________   \n",
       "⎮   ╱  x        \n",
       "⎮ ╲╱  ℯ  + 4  dx\n",
       "⌡               \n",
       "0               "
      ]
     },
     "execution_count": 72,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "I5.doit()"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 73,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "image/png": "iVBORw0KGgoAAAANSUhEUgAAAJ8AAAAPBAMAAAAIUwCQAAAAMFBMVEX///8AAAAAAAAAAAAAAAAA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\nfiOWBhMAAAAASUVORK5CYII=\n",
      "text/latex": [
       "$$2.38910363799198$$"
      ],
      "text/plain": [
       "2.38910363799198"
      ]
     },
     "execution_count": 73,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "I5.n()"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Vergleich mit der Methode aus der CompLA (nicht prüfungsrelevant für CompAna)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 74,
   "metadata": {
    "collapsed": true
   },
   "outputs": [],
   "source": [
    "import numpy as np\n",
    "from scipy import integrate"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 75,
   "metadata": {
    "collapsed": true
   },
   "outputs": [],
   "source": [
    "def f(x):\n",
    "    return np.sqrt(np.exp(x)+4)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 76,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "image/png": "iVBORw0KGgoAAAANSUhEUgAAALMAAAAPBAMAAABdMuPOAAAAMFBMVEX///8AAAAAAAAAAAAAAAAA\nAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAv3aB7AAAAD3RSTlMAIpm7MhCriUTv3c12\nVGZoascqAAAACXBIWXMAAA7EAAAOxAGVKw4bAAADCUlEQVQ4EbWTz29UVRTHP9Pp60znt24IuhnH\ngCZuJhCNG9PZmbjpCwsSYDGlUQQS7aQQJiqxE3fGBSWKKZCIunHJLJSVobMgGgKJL0Rj3DCPjYRo\naKsWgdqO33tO/RO8i5nvnfl+P++8c88l09jbQivTPZEQ3ZzvET13qhZ+QepuC3Z0X5Huft9y9dps\nt9txz77u4WDTMmVp3zNeh/3kHobtk5S3GIO3OJBmzkL5HBST6AsydYYph3pjfVcro9FoxjzRZ+xU\nEb/EuLK0BaHQhDfgvMj8kPCY2/A0x2CW/bvX4R68R2mT6ufsIdt09QxMuGdikYl6NDsXY8rTFoTr\nQl+AuZbQH/aiDU7AQTZgAUpCvwTDOL9Eu59fk8UUHTjpnmqf3KqKinFlaQuS+U3o5djRasEWZ17k\n/ehPuVvmeABnEjGHaXZRX6b0kR+4p92k9JehXYW010SuKLTW5di+DnaYGN1NuaiqewEd/SH0QKf5\nFdWvTz8vk5RWFvdMqmqdk6p2ZWmv+o6j838HPzu+1BMWNuLQjeU0ODL6Y7pD5t1dtK8ymZiSU6WZ\nR90obhl6W4W0oaOBowszgQy5JcpHLy9RHJQXEkOraqHhnV57nTGNjJRKX9XsmOdbft40NKYsbegc\njm4oYut8/CulRzG/X1toBcd/DWHynGakoneXUll9CfNUbr7qvcaUp8P5/+joUt24T8BUonmeTrTV\nPAWHjnGYZGpkNwt9KlumfGTdozGUS73WkvJ0qKnj6FsoAqOYqZ9WZUm1ueIt2wsrcXVdaA1u5aEp\naM/IYR41RtrRxZnQKKWFLh85MvfpINzIUkB/rEmp6bnZ2r3a+JqjdWVeYLJJYW1cvV40pfeq6zIF\nT+Us04oK7crS9rrqXBOe6s4f153gPuV/+EiN5kZ8KHV0IYk+oZIyHPAmBxJXDIU2T+mb6KqKEtqV\npbfRVaGXR6PHaEjyjWd7jM/O98g1NMJjxzbeJjr9XQs+aLys8Wm8vq3YmYRtGPP5Rg92X7qSurK0\nBclNPUhl+L/Wv5fONEJW6wDmAAAAAElFTkSuQmCC\n",
      "text/latex": [
       "$$2.3891036379919814$$"
      ],
      "text/plain": [
       "2.3891036379919814"
      ]
     },
     "execution_count": 76,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "res, err = integrate.quad(f, 0, 1)\n",
    "res"
   ]
  }
 ],
 "metadata": {
  "anaconda-cloud": {},
  "kernelspec": {
   "display_name": "Python [conda root]",
   "language": "python",
   "name": "conda-root-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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}