abhigyan/Jupyter/Introduction to SageMath.ipynb

{
 "cells": [
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "# Symbolic Computation using SageMath\n",
    "The most basic data type of SageMath is an expression"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 1,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "(x, y)"
      ]
     },
     "execution_count": 1,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "var('x,y')"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 2,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "<class 'sage.symbolic.expression.Expression'>"
      ]
     },
     "execution_count": 2,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "type(x)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 3,
   "metadata": {},
   "outputs": [],
   "source": [
    "poly1  = (x+3)^3"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 4,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/html": [
       "<html><script type=\"math/tex; mode=display\">\\newcommand{\\Bold}[1]{\\mathbf{#1}}{\\left(x + 3\\right)}^{3}</script></html>"
      ],
      "text/plain": [
       "(x + 3)^3"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "poly1.show()"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 5,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/html": [
       "<html><script type=\"math/tex; mode=display\">\\newcommand{\\Bold}[1]{\\mathbf{#1}}x^{3} + 9 \\, x^{2} + 27 \\, x + 27</script></html>"
      ],
      "text/plain": [
       "x^3 + 9*x^2 + 27*x + 27"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "poly1.expand().show()"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 6,
   "metadata": {},
   "outputs": [],
   "source": [
    "poly2 = x^2+2*x-3"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 7,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/html": [
       "<html><script type=\"math/tex; mode=display\">\\newcommand{\\Bold}[1]{\\mathbf{#1}}{\\left(x + 3\\right)} {\\left(x - 1\\right)}</script></html>"
      ],
      "text/plain": [
       "(x + 3)*(x - 1)"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "poly2.factor().show()"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Now, we will write an expression with both an LHS and an RHS using the == symbol"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 8,
   "metadata": {},
   "outputs": [],
   "source": [
    "expr1 = (y-3)/(x^2+1)==x^3-5"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 9,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/html": [
       "<html><script type=\"math/tex; mode=display\">\\newcommand{\\Bold}[1]{\\mathbf{#1}}\\frac{y - 3}{x^{2} + 1} = x^{3} - 5</script></html>"
      ],
      "text/plain": [
       "(y - 3)/(x^2 + 1) == x^3 - 5"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "expr1.show()"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Now, we will access each side of the equation:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 10,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "LHS:\n"
     ]
    },
    {
     "data": {
      "text/html": [
       "<html><script type=\"math/tex; mode=display\">\\newcommand{\\Bold}[1]{\\mathbf{#1}}\\frac{y - 3}{x^{2} + 1}</script></html>"
      ],
      "text/plain": [
       "(y - 3)/(x^2 + 1)"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    },
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "RHS: \n"
     ]
    },
    {
     "data": {
      "text/html": [
       "<html><script type=\"math/tex; mode=display\">\\newcommand{\\Bold}[1]{\\mathbf{#1}}x^{3} - 5</script></html>"
      ],
      "text/plain": [
       "x^3 - 5"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "print(\"LHS:\" )\n",
    "expr1.lhs().show()\n",
    "print(\"RHS: \")\n",
    "expr1.rhs().show()"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Manipulating both sides of the equation"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 11,
   "metadata": {},
   "outputs": [],
   "source": [
    "expr2 = expr1.multiply_both_sides(x^2+1)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 12,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/html": [
       "<html><script type=\"math/tex; mode=display\">\\newcommand{\\Bold}[1]{\\mathbf{#1}}y - 3 = {\\left(x^{3} - 5\\right)} {\\left(x^{2} + 1\\right)}</script></html>"
      ],
      "text/plain": [
       "y - 3 == (x^3 - 5)*(x^2 + 1)"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "expr2.show()"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 13,
   "metadata": {},
   "outputs": [],
   "source": [
    "expr3 =  expr2.add_to_both_sides(3)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 14,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/html": [
       "<html><script type=\"math/tex; mode=display\">\\newcommand{\\Bold}[1]{\\mathbf{#1}}y = {\\left(x^{3} - 5\\right)} {\\left(x^{2} + 1\\right)} + 3</script></html>"
      ],
      "text/plain": [
       "y == (x^3 - 5)*(x^2 + 1) + 3"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "expr3.show()"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 52,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "(a, b, c, d)"
      ]
     },
     "execution_count": 52,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "var('a,b,c,d')"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 16,
   "metadata": {},
   "outputs": [],
   "source": [
    "[x1,x2]= solve(a*x^2+b*x+c==0,x)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 17,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/html": [
       "<html><script type=\"math/tex; mode=display\">\\newcommand{\\Bold}[1]{\\mathbf{#1}}x = -\\frac{b + \\sqrt{b^{2} - 4 \\, a c}}{2 \\, a}</script></html>"
      ],
      "text/plain": [
       "x == -1/2*(b + sqrt(b^2 - 4*a*c))/a"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    },
    {
     "data": {
      "text/html": [
       "<html><script type=\"math/tex; mode=display\">\\newcommand{\\Bold}[1]{\\mathbf{#1}}x = -\\frac{b - \\sqrt{b^{2} - 4 \\, a c}}{2 \\, a}</script></html>"
      ],
      "text/plain": [
       "x == -1/2*(b - sqrt(b^2 - 4*a*c))/a"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "x1.show()\n",
    "x2.show()"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Greatest Common Factor of Expressions"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 18,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/html": [
       "<html><script type=\"math/tex; mode=display\">\\newcommand{\\Bold}[1]{\\mathbf{#1}}{\\left(x - 3\\right)} {\\left(x - 5\\right)}</script></html>"
      ],
      "text/plain": [
       "(x - 3)*(x - 5)"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    },
    {
     "data": {
      "text/html": [
       "<html><script type=\"math/tex; mode=display\">\\newcommand{\\Bold}[1]{\\mathbf{#1}}{\\left(x - 2\\right)} {\\left(x - 3\\right)}</script></html>"
      ],
      "text/plain": [
       "(x - 2)*(x - 3)"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    },
    {
     "data": {
      "text/html": [
       "<html><script type=\"math/tex; mode=display\">\\newcommand{\\Bold}[1]{\\mathbf{#1}}x - 3</script></html>"
      ],
      "text/plain": [
       "x - 3"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "gcdpart1 = x^2-8*x+15\n",
    "gcdpart2 = x^2-5*x+6\n",
    "factor(gcdpart1).show()\n",
    "factor(gcdpart2).show()\n",
    "gcd(gcdpart1,gcdpart2).show()"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Partial Fractions:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 19,
   "metadata": {},
   "outputs": [],
   "source": [
    "expr5 = (x+1)/((x^2)*(x+2)^3)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 20,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/html": [
       "<html><script type=\"math/tex; mode=display\">\\newcommand{\\Bold}[1]{\\mathbf{#1}}\\frac{x + 1}{{\\left(x + 2\\right)}^{3} x^{2}}</script></html>"
      ],
      "text/plain": [
       "(x + 1)/((x + 2)^3*x^2)"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "expr5.show()"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 21,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/html": [
       "<html><script type=\"math/tex; mode=display\">\\newcommand{\\Bold}[1]{\\mathbf{#1}}{\\left(x + 2\\right)}^{3} x^{2}</script></html>"
      ],
      "text/plain": [
       "(x + 2)^3*x^2"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "expr5.denominator().show()"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 22,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/html": [
       "<html><script type=\"math/tex; mode=display\">\\newcommand{\\Bold}[1]{\\mathbf{#1}}\\frac{1}{{\\left(x^{3} + 6 \\, x^{2} + 12 \\, x + 8\\right)} x} + \\frac{1}{{\\left(x^{3} + 6 \\, x^{2} + 12 \\, x + 8\\right)} x^{2}}</script></html>"
      ],
      "text/plain": [
       "1/((x^3 + 6*x^2 + 12*x + 8)*x) + 1/((x^3 + 6*x^2 + 12*x + 8)*x^2)"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "expr5.expand().show()"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 23,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/html": [
       "<html><script type=\"math/tex; mode=display\">\\newcommand{\\Bold}[1]{\\mathbf{#1}}\\frac{1}{16 \\, {\\left(x + 2\\right)}} - \\frac{1}{16 \\, x} + \\frac{1}{8 \\, x^{2}} - \\frac{1}{4 \\, {\\left(x + 2\\right)}^{3}}</script></html>"
      ],
      "text/plain": [
       "1/16/(x + 2) - 1/16/x + 1/8/x^2 - 1/4/(x + 2)^3"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "expr5.partial_fraction().show()"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Reducing Trigonometric Expressions"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 24,
   "metadata": {},
   "outputs": [],
   "source": [
    "texp1 = sin(x)^2*cos(x)^2+sin(y)^2"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 25,
   "metadata": {},
   "outputs": [],
   "source": [
    "texp2 = texp1.trig_reduce()"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 26,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/html": [
       "<html><script type=\"math/tex; mode=display\">\\newcommand{\\Bold}[1]{\\mathbf{#1}}-\\frac{1}{8} \\, \\cos\\left(4 \\, x\\right) - \\frac{1}{2} \\, \\cos\\left(2 \\, y\\right) + \\frac{5}{8}</script></html>"
      ],
      "text/plain": [
       "-1/8*cos(4*x) - 1/2*cos(2*y) + 5/8"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "texp2.show()"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 27,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/html": [
       "<html><script type=\"math/tex; mode=display\">\\newcommand{\\Bold}[1]{\\mathbf{#1}}-\\cos\\left(x\\right)^{4} + \\cos\\left(x\\right)^{2} + \\sin\\left(y\\right)^{2}</script></html>"
      ],
      "text/plain": [
       "-cos(x)^4 + cos(x)^2 + sin(y)^2"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "texp2.trig_simplify().show()"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## General Expressions:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 28,
   "metadata": {},
   "outputs": [],
   "source": [
    "gexp1 = log(x^2*sin(x)/sqrt(x+1))"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 29,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/html": [
       "<html><script type=\"math/tex; mode=display\">\\newcommand{\\Bold}[1]{\\mathbf{#1}}\\log\\left(\\frac{x^{2} \\sin\\left(x\\right)}{\\sqrt{x + 1}}\\right)</script></html>"
      ],
      "text/plain": [
       "log(x^2*sin(x)/sqrt(x + 1))"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "gexp1.show()"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 30,
   "metadata": {},
   "outputs": [],
   "source": [
    "gexp2 = gexp1.expand_log()"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 31,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/html": [
       "<html><script type=\"math/tex; mode=display\">\\newcommand{\\Bold}[1]{\\mathbf{#1}}-\\frac{1}{2} \\, \\log\\left(x + 1\\right) + 2 \\, \\log\\left(x\\right) + \\log\\left(\\sin\\left(x\\right)\\right)</script></html>"
      ],
      "text/plain": [
       "-1/2*log(x + 1) + 2*log(x) + log(sin(x))"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "gexp2.show()"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 32,
   "metadata": {},
   "outputs": [],
   "source": [
    "gexp3 = gexp2.simplify_log()"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 33,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/html": [
       "<html><script type=\"math/tex; mode=display\">\\newcommand{\\Bold}[1]{\\mathbf{#1}}\\log\\left(x^{2} \\sin\\left(x\\right)\\right) - \\frac{1}{2} \\, \\log\\left(x + 1\\right)</script></html>"
      ],
      "text/plain": [
       "log(x^2*sin(x)) - 1/2*log(x + 1)"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "gexp3.show()"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Rational Expressions"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 34,
   "metadata": {},
   "outputs": [],
   "source": [
    "rexp1 = (x+1)/(x^2+x)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 35,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/html": [
       "<html><script type=\"math/tex; mode=display\">\\newcommand{\\Bold}[1]{\\mathbf{#1}}\\frac{x + 1}{x^{2} + x}</script></html>"
      ],
      "text/plain": [
       "(x + 1)/(x^2 + x)"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "rexp1.show()"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 36,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/html": [
       "<html><script type=\"math/tex; mode=display\">\\newcommand{\\Bold}[1]{\\mathbf{#1}}\\frac{1}{x}</script></html>"
      ],
      "text/plain": [
       "1/x"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "rexp1.simplify_rational().show()"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Learning Differential Geometry becomes easy with SageMath :)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Differentiating\n",
    "### Single- and Multi-Variable"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 37,
   "metadata": {},
   "outputs": [],
   "source": [
    "f(x) = a*x^2+b*x+c"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 38,
   "metadata": {},
   "outputs": [],
   "source": [
    "df(x)=f.derivative(x)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 39,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/html": [
       "<html><script type=\"math/tex; mode=display\">\\newcommand{\\Bold}[1]{\\mathbf{#1}}2 \\, a x + b</script></html>"
      ],
      "text/plain": [
       "2*a*x + b"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "df(x).show()"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 40,
   "metadata": {},
   "outputs": [],
   "source": [
    "f1(x,y) = a*x^2+b*y^2+2*c*x*y"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 41,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/html": [
       "<html><script type=\"math/tex; mode=display\">\\newcommand{\\Bold}[1]{\\mathbf{#1}}a x^{2} + 2 \\, c x y + b y^{2}</script></html>"
      ],
      "text/plain": [
       "a*x^2 + 2*c*x*y + b*y^2"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    },
    {
     "data": {
      "text/html": [
       "<html><script type=\"math/tex; mode=display\">\\newcommand{\\Bold}[1]{\\mathbf{#1}}\\left( x, y \\right) \\ {\\mapsto} \\ 2 \\, c x + 2 \\, b y</script></html>"
      ],
      "text/plain": [
       "(x, y) |--> 2*c*x + 2*b*y"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    },
    {
     "data": {
      "text/html": [
       "<html><script type=\"math/tex; mode=display\">\\newcommand{\\Bold}[1]{\\mathbf{#1}}\\left( x, y \\right) \\ {\\mapsto} \\ 2 \\, a x + 2 \\, c y</script></html>"
      ],
      "text/plain": [
       "(x, y) |--> 2*a*x + 2*c*y"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "f1(x,y).show()\n",
    "\n",
    "df1dy(x,y) = diff(f1,y)\n",
    "df1dx(x,y) = diff(f1,x)\n",
    "\n",
    "df1dy.show()\n",
    "df1dx.show()"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "( learn Risch Algorithm )"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Integral Calculus\n",
    "\n",
    "### Indefinite Integral"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 42,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/html": [
       "<html><script type=\"math/tex; mode=display\">\\newcommand{\\Bold}[1]{\\mathbf{#1}}\\frac{1}{2} \\, \\sqrt{\\pi} \\operatorname{erf}\\left(x\\right)</script></html>"
      ],
      "text/plain": [
       "1/2*sqrt(pi)*erf(x)"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "integral(exp(-x^2),x).show()"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 43,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "0"
      ]
     },
     "execution_count": 43,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "erf(0)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 44,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/html": [
       "<html><script type=\"math/tex; mode=display\">\\newcommand{\\Bold}[1]{\\mathbf{#1}}\\operatorname{erf}\\left(x\\right)</script></html>"
      ],
      "text/plain": [
       "erf(x)"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "erf(x).expand().show()"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 45,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/html": [
       "<html><script type=\"math/tex; mode=display\">\\newcommand{\\Bold}[1]{\\mathbf{#1}}\\arctan\\left(x\\right)</script></html>"
      ],
      "text/plain": [
       "arctan(x)"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "integral(1/(1+x^2),x).show()"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### Definite Integral"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 46,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/html": [
       "<html><script type=\"math/tex; mode=display\">\\newcommand{\\Bold}[1]{\\mathbf{#1}}\\frac{1}{4} \\, \\pi</script></html>"
      ],
      "text/plain": [
       "1/4*pi"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "integral(1/(1+x^2),x,0,1).show()"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Plotting"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 47,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": 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ZrrsOLr8c5syBX/866qqUypyKlSQpDsaNg8suC7cBGz8eOnWKuiKlA0fsJKWsWCxGbm4ueXl5UZeiNLJmDfTsCaefHu71+u67hjolTkYQBEHURUhSPJWUlJCdnU1xcTFZWVlRl6MU9sEH0KULfPwxDB0K3btDRkbUVSmdOGInSdJOCgJ4+GFo1Src8/Wtt6BHD0OdEs9gJ0nSTiguhvPOgyuuCPd7nT0bDjss6qqUrlw8IUnSDpo1C84/P9webPTocIswKUqO2EmStJ0qKuDOO6F9e8jJCfd5NdSpOjDYSZK0HYqK4MQToW/fsNHw9Olw4IFRVyWFnIqVJGkbTZwIF10U7hrxyivQsWPUFUkbcsROkqStKC+H226Dk04Kd46YN89Qp+rJETtJkrZgyRLo2hWmTYPbb4c+fdznVdWXwU6SpM149VW44IJwW7BJk+CYY6KuSNoy/80hSdJPlJdD//6Qnw+/+lU49WqoUzJwxE6SpB8pKgpH6aZMgYED4Q9/CEfspGRgsJOUsmKxGLFYjPLy8qhLUZKYNCm8ny4jI5yGPe64qCuStk9GEARB1EVIUjyVlJSQnZ1NcXExWVlZUZejaqi8PFwYMXAgHH88jBoVNh6Wko0jdpKktPaf/4SjdK+9Ft5Xd+utTr0qeRnsJElpa9YsOOccKC2FwkI44YSoK5J2jqtiJUlpJwggFoPf/AYaN4a5cw11Sg0GO0lSWlm1KtwW7JproHv3cPVro0ZRVyVVDadiJUlp4+OP4eyzYeFCePppOO+8qCuSqpYjdpKktDB2LLRuDevWwZtvGuqUmgx2kqSUtn499O4djtSdeCK89RY0bx51VVJ8OBUrSUpZRUXhyNyMGXD33XDDDWHzYSlVOWInKeGGDRvGgQceSO3atWnVqhXTp0/fpvNef/11dtllF37961/HuUKlgtdfh5Yt4aOPYPJkuPFGQ51Sn8FOUkKNHj2aXr160a9fP+bOnUv79u3p1KkTixYt2uJ5xcXFdOvWjRPsSaGtCAK491449lg46CD4v/+D9u2jrkpKDLcUk5RQbdq0oWXLlgwfPrzyWLNmzTjjjDMoKCjY7HnnnXceTZs2pWbNmrz44ovMmzdvm6/plmLpY9UquPxyeOYZ+P3voaAAdt016qqkxHHETlLClJWVMWfOHPLz8zc4np+fz8yZMzd73mOPPcZnn31G//79t+k6paWllJSUbPBQ6lu4ENq1g3/+E559Fu66y1Cn9GOwk5Qwy5Yto7y8nJyf7K6ek5NDUVHRJs/55JNP6NOnD6NGjWKXXbZtvVdBQQHZ2dmVj8aNG+907areXn01bGWyatX/tgmT0pHBTlLCZfzkDvYgCDY6BlBeXk7Xrl0ZOHAgBx988Da/ft++fSkuLq58LF68eKdrVvUUBOHI3IknQl4ezJ4NLVpEXZUUHdudSEqY+vXrU7NmzY1G55YuXbrRKB7AypUrefvtt5k7dy7XXHMNABUVFQRBwC677MLEiRM5/vjjNzovMzOTzMzM+LwJVRurV4f30z39NPTpA7ffDjVrRl2VFC2DnaSEqVWrFq1ataKwsJAzzzyz8nhhYSGnn376Rs/Pysrivffe2+DYsGHDmDRpEs8//zwHHnhg3GtW9bRwIZx5JnzySXg/nVOvUshgJymhbrzxRi666CJat25N27ZtGTFiBIsWLeLqq68GwmnUr776iieeeIIaNWrQ4ifzavvssw+1a9fe6LjSx6uvQpcusMce4f10hx4adUVS9WGwk5RQXbp0Yfny5QwaNIglS5bQokULxo8fT5MmTQBYsmTJVnvaKT0FAQwdGm4P1rEjPPUU7LVX1FVJ1Yt97CSlPPvYJT/vp5O2jSN2kqRq7cf3040eDeeeG3VFUvVlsJMkVVuTJ4cLI7Kz4Y034LDDoq5Iqt7sYydJqpaGD4f8fDj88LA/naFO2jqDnSSpWlm3Dnr2hB49wsfLL7tIQtpWTsVKkqqN5cvDqdfp02HECLjiiqgrkpKLwU6SVC3Mnw+nnQbFxfDaa/Cb30RdkZR8nIqVlLJisRi5ubnk5eVFXYq2Yvx4OPJIqFMH3nrLUCftKPvYSUp59rGrvoIA7r47bDrcuTM8+STUqxd1VVLycsROkhSJtWvhkkvg5pvDpsMvvGCok3aW99hJkhKuqChsOjx3LowaBV27Rl2RlBoMdpKkhJo7N1wkUV4O06bBEUdEXZGUOpyKlSQlzHPPwVFHQYMGYdNhQ51UtQx2kqS4q6iAgQPDfV5PPz0cqWvUKOqqpNTjVKwkKa7WrAkXSTz7LPz5z9C3L2RkRF2VlJoMdpKkuPnmm3CE7t13YcwYOOusqCuSUpvBTpIUF//+N5x6KpSVhVOvrVtHXZGU+rzHTpJU5SZMgHbtYI894M03DXVSohjsJElVatgwOOUUOOYYmDEDGjeOuiIpfRjsJElVorwcevWCnj3huuvgxRdh992jrkpKL95jJ0naaStXwvnnh1Oww4ZB9+5RVySlJ4OdpJQVi8WIxWKUl5dHXUpKW7QIOneGzz+Hl16CE0+MuiIpfWUEQRBEXYQkxVNJSQnZ2dkUFxeTlZUVdTkpZfbscHuwzMww1DVvHnVFUnrzHjtJ0g4ZOzZcIHHAAeHKV0OdFD2DnSRpuwQB3HknnH12OFo3aRLk5ERdlSQw2EmStsO6dXD55dCnD9x6Kzz1FNSpE3VVkn7g4glJ0jYpLobf/hamToXHH4du3aKuSNJPGewkSVv15Zdw8snhCthXXoHjjou6IkmbYrCTJG3RvHnhThK77AKvv+4iCak68x47SdJmvfIKtG8PDRrArFmGOqm6M9hJkjbpkUf+t+fr1KnQsGHUFUnaGoOdJGkDQRCueL38crjiCvd8lZKJ99hJkiqVlcFll8GTT4a96m6+GTIyoq5K0rYy2EmSAPj2WzjrLJg5E555Brp0iboiSdvLYCdJ4osvoFMnKCqCV18NF0xISj7eYycpZcViMXJzc8nLy4u6lGptzhw48khYuxbeeMNQJyWzjCAIgqiLkKR4KikpITs7m+LiYrKysqIup1p56SU491xo0QL++U/YZ5+oK5K0Mxyxk6Q0NXw4nHYadOwIkycb6qRUYLCTpDRTUQG33AI9esA118CYMVC3btRVSaoKLp6QpDRSVgaXXgpPPw1Dh8INN0RdkaSqZLCTpDRRUhK2M5k+PWxncu65UVckqaoZ7CQpDSxZErYz+fxzmDgx3CZMUuox2ElSivvoIzjxRFi/HmbMCFfASkpNLp6QpBT2xhvQrh3stlv4a0OdlNoMdpISbtiwYRx44IHUrl2bVq1aMX369M0+d+zYsXTs2JG9996brKws2rZtyyuvvJLAapPXuHFw/PHQvHk4Ute4cdQVSYo3g52khBo9ejS9evWiX79+zJ07l/bt29OpUycWLVq0yedPmzaNjh07Mn78eObMmcNxxx1H586dmTt3boIrTy4jRsCZZ8Ipp4T31O25Z9QVSUoEd56QlFBt2rShZcuWDB8+vPJYs2bNOOOMMygoKNim12jevDldunThj3/84zY9P512nggCGDAABg0Ke9Tdcw/UrBl1VZISxRE7SQlTVlbGnDlzyM/P3+B4fn4+M2fO3KbXqKioYOXKley1116bfU5paSklJSUbPNLB+vVwxRVhqLvjDvjb3wx1Urox2ElKmGXLllFeXk5OTs4Gx3NycigqKtqm17j77rtZtWoV526hCVtBQQHZ2dmVj8ZpcHPZqlVwxhnw+OPwxBPhzhIZGVFXJSnRDHaSEi7jJ4kjCIKNjm3K008/zYABAxg9ejT7bGFj0759+1JcXFz5WLx48U7XXJ395z/hIokpU+Cll+Cii6KuSFJU7GMnKWHq169PzZo1NxqdW7p06UajeD81evRoLrvsMp577jk6dOiwxedmZmaSmZm50/UmgwUL4KSToLgYpk6FVq2irkhSlByxk5QwtWrVolWrVhQWFm5wvLCwkHbt2m32vKeffppLLrmEp556ilNOOSXeZSaNOXOgbdvw12+8YaiT5IidpAS78cYbueiii2jdujVt27ZlxIgRLFq0iKuvvhoIp1G/+uornnjiCSAMdd26dePee+/lyCOPrBztq1OnDtnZ2ZG9j6i9+mrYziQ3F/71L9h776grklQdOGInKaG6dOnCPffcw6BBg/j1r3/NtGnTGD9+PE2aNAFgyZIlG/S0e/DBB1m/fj09e/akYcOGlY/rr78+qrcQuWefhZNPhqOPhkmTDHWS/sc+dpJSXir1sYvF4NproWtXeOwx2HXXqCuSVJ04YidJSSAIoH//sOlwr15hSxNDnaSf8h47SarmysuhZ0948MGw8XDv3vaok7RpBjtJqsbWroULL4QXXoBHHoHf/S7qiiRVZwY7SaqmSkrg9NNh1qww2J12WtQVSaruDHaSVA198w106hQ2IJ44Edq3j7oiScnAYCdJ1cyCBZCfD6tXw/TpcOihUVckKVm4KlaSqpF586BdO6hRA2bONNRJ2j4GO0kpKxaLkZubS15eXtSlbJMpU+CYY2C//WDGDDjggKgrkpRsbFAsKeUlQ4PisWPDpsNHHx0ulKhXL+qKJCUjR+wkKWIPPQTnnBOugH3pJUOdpB1nsJOkiAQB3H47XHkldO8OTz0FmZlRVyUpmbkqVpIiUFEB118P998PAwfCbbe5m4SknWewk6QEKyuDiy+G0aNh+HC4+uqoK5KUKgx2kpRA338PZ58droB97rnw15JUVQx2kpQgy5bBySfDhx/ChAlw3HFRVyQp1RjsJCkBvvwy3E1i2bJwtK5ly6grkpSKDHaSFGeffAIdOoS/njEDDj442nokpS7bnUhSHM2bFzYdrlvXUCcp/gx2khQnM2aEW4Q1bgzTp4f/laR4MthJUhyMHx/eU9eyJUyaBPXrR12RpHRgsJOUsmKxGLm5ueTl5SX0uk8/HW4P1rEjvPwyVNPtaSWloIwgCIKoi5CkeCopKSE7O5vi4mKy4pyyhg+Hnj3hoovgkUdgF5eoSUogR+wkqQoEAfz5z9CjB1x7LTz2mKFOUuIZ7CRpJwUB3HQT3HpruO/rPfdADb9dJUXAf09K0k5Yvx6uvDIcofvb38LROkmKisFOknbQ2rXQtSuMGwd//ztceGHUFUlKdwY7SdoBK1fCmWfC66/DCy9A585RVyRJBjtJ2m7Ll8PJJ8MHH8CECWETYkmqDgx2krQdvvoqbDy8dClMmRI2IJak6sJgJ0nb6NNPw6bD5eXhFmGHHBJ1RZK0IRfkS9I2eOcdOPpoyMwM94A11Emqjgx2krQVr78e3kfXqFE4Urf//lFXJEmbZrCTpC2YMCGcfv31r2HyZNh776grkqTNM9hJ0maMHh22MenQAV5+GeK8zawk7TSDnaSUFYvFyM3NJS8vb7vPffBBOP98OO88GDMG6tSJQ4GSVMUygiAIoi5CkuKppKSE7OxsiouLydrKsFsQwB13wB/+EG4P5r6vkpKJX1eS9F9BAL17h6Guf3+4915DnaTkYh87SSLsTXfVVfDII+Eo3fXXR12RJG0/g52ktFdaChdcAC++CI8/Dt26RV2RJO0Yg52ktPb993DmmWF/urFj4bTToq5IknacwU5S2lqxAk4+GebPD/vVHXts1BVJ0s4x2ElKS19/Dfn5UFQEkyZB69ZRVyRJO89gJyntfPZZuJvEunXhFGyzZlFXJElVw4X8ktLKu+/C0UfDLrvAjBmGOkmpxWAnKW28+SYccww0aBCGuiZNoq5IkqqWwU5S2jj9dDj0UJgyBfbZJ+pqJKnq7dA9dkEQsHLlyqquRZKqRGlpKaWlpZU/P/fcGgDati1h1CjIyICSkqiqk6RtV69ePTIyMrb5+Tu0V+wP+y5KkiQpfrZlj+sf26Fgtz0jdiUlJTRu3JjFixdvV2E7Ky8vj9mzZyfsel4zfvwMec3t9cOIXSxWizvuqM1vf7uE558/hPnz59OoUaO4XfenUvHPNl2v6feQ19xZO/oZ2t4Rux2ais3IyNjuD3ZWVlZC/89Qs2bNhF7Pa8afnyGvua2CAPr0gSFD4Lbb4IorSnj++fAL0s+Q19wZfg95zZ0V789Qyvax69mzp9dMoWtGIV1FxO0mAAAgAElEQVT+bFPtmuXl0L07PPQQ/PWv0KsXfPll3C63Ran2Z5vu14xCuvzZpss1E2GHpmK3xw/3423vHLH0Az9D2lalpXDRRTBmDDzyCFxySXj8yy+/rJwC2W+//SKtUcnJ7yHtrER9huI+YpeZmUn//v3JzMyM96WUovwMaVusWgVnnRW2MhkzBs4443+/98Nnx8+QdpTfQ9pZifoMxX3ETpLibcUKOPVUeO89+Mc/4PjjN/x9R1skpYuUvcdOUnpYsgTy88P/TpoEeXlRVyRJ0THYSUpaCxZAx47hvXXTpkFubtQVSVK03FJMUlJ67z04+mioWRNef91QJ0lgsJOUhGbNgmOOgZwcmD4dmjSJuiJJqh7iHuzGjh3LiSeeSP369cnIyGDevHnxvqSSUBAEDBgwgH333Zc6depw7LHH8v7772/xnJEjR5KRkbHRY+3atQmqWlEoLIQTToDmzcMVsDk5UVekVDFs2DAOPPBAateuTatWrZg+ffpmnztlypRNfv98+OGHCaxYyWDatGl07tyZfffdl4yMDF588cW4Xi/uwW7VqlUcddRR3HHHHfG+lJLYkCFDGDp0KPfffz+zZ8+mQYMGdOzYcatb12VlZbFkyZINHrVr105Q1Uq055+HU06BY4+FV14Bt6xWVRk9ejS9evWiX79+zJ07l/bt29OpUycWLVq0xfM++uijDb5/mjZtmqCKlSxWrVrFr371K+6///7EXDBIkIULFwZAMHfu3ERdUkmioqIiaNCgQXDHHXdUHlu7dm2QnZ0dPPDAA5s977HHHguys7MTUaKqgREjgqBGjSDo2jUIysq279zi4uIACIqLi+NTnJLeEUccEVx99dUbHDvkkEOCPn36bPL5kydPDoDg22+/TUR5ShFA8MILL8T1Gt5jp8gtXLiQoqIi8vPzK49lZmZyzDHHMHPmzC2e+/3339OkSRP2228/Tj31VObOnRvvchWBIUPgyivDrcL+/nfYdddtOy8Wi5Gbm0uePVC0BWVlZcyZM2eD7yCA/Pz8rX4HHX744TRs2JATTjiByZMnx7NMaZsY7BS5oqIiAHJ+crNUTk5O5e9tyiGHHMLIkSMZN24cTz/9NLVr1+aoo47ik08+iWu9SpwggD594JZb4Lbb4L77oMZ2fGv17NmT+fPnM3v27PgVqaS3bNkyysvLt+s7qGHDhowYMYIxY8YwduxYfvnLX3LCCScwbdq0RJQsbVaVBrtRo0ax++67Vz62dOOp0tdPPyfr1q0DICMjY4PnBUGw0bEfO/LII7nwwgv51a9+Rfv27Xn22Wc5+OCDue++++JavxKjvByuugruvBP++lcYNAi28HGQdtr2fAf98pe/5IorrqBly5a0bduWYcOGccopp3DXXXclolRps6q0QfFpp51GmzZtKn9u1KhRVb68UsRPPyelpaVAOHLXsGHDyuNLly7d6F/QW1KjRg3y8vIcsUsBpaVw0UUwdiyMHAkXXxx1RUpl9evXp2bNmhuNzm3vd9CRRx7Jk08+WdXlSdulSoNdvXr1qFevXlW+pFLQTz8nQRDQoEEDCgsLOfzww4HwnpepU6dy5513bvPrBkHAvHnzOPTQQ6u8ZiXOqlVw1lkwdSqMGQOnnx51RUp1tWrVolWrVhQWFnLmmWdWHi8sLOT07fgAzp07d4N/nEpRiPuWYitWrGDRokV8/fXXQLg0HKBBgwY0aNAg3pdXEsjIyKBXr14MHjyYpk2b0rRpUwYPHkzdunXp2rVr5fO6detGo0aNKCgoAGDgwIEceeSRNG3alJKSEv72t78xb948YrFYVG9FO2nFCjj11HBXiZdfhuOOi7oipYsbb7yRiy66iNatW9O2bVtGjBjBokWLuPrqqwHo27cvX331FU888QQA99xzDwcccADNmzenrKyMJ598kjFjxjBmzJgo34aqoe+//55PP/208ueFCxcyb9489tprL/bff/8qv17cg924ceO49NJLK38+77zzAOjfvz8DBgyI9+WVJHr37s2aNWvo0aMH3377LW3atGHixIkbjOwtWrSIGj+6c/67777jyiuvpKioiOzsbA4//HCmTZvGEUccEcVb0E5asgTy88P/TpoELmRVInXp0oXly5czaNAglixZQosWLRg/fjxN/rutyZIlSzboaVdWVsZNN93EV199RZ06dWjevDkvvfQSJ598clRvQdXU22+/zXE/+lfqjTfeCMDFF1/MyJEjq/x6Gf/tqyJJkVmwADp2DO+tKyyEZs2q9vVLSkrIzs6muLiYrKysqn1xSapGbHciKVLvvQdHHw01a8Lrr1d9qJOkdGKwkxSZWbPgmGPC/V6nT4f/znpJknaQwU5SJAoL4YQToEULmDIlDHeSpJ1jsJOUcM8/D6ecEq56nTABsrOjrkiSUoPBTlJCPfQQdOkC55wDL7wAdetGXZEkpQ6DnaSEGTIErrwSuneHv/8ddt01vteLxWLk5uaSZ+8USWnCdieS4i4IoE+fMNjddhsMHJjYfV9tdyIpXcS9QbGk9LZ+fThK99hj8Ne/Qq9eUVckSanLYCcpbtasgfPOg/Hj4ckn4YILoq5IklKbwU5SXHz3HZx2Grz9NowbB506RV2RJKU+g52kKrdkCZx0EixeDK+9Bm3bRl2RJKUHg52kKvXZZ5CfH+77On06NG8edUWSlD5sdyKpysybB0cdBbvsEu77aqiTpMQy2EmqElOnhvu+Nm4MM2a476skRcFgJ2mnvfginHgi5OXBpEmw995RVyRJ6clgJ2mnPPIInH12uAL2pZegXr2oK5Kk9GWwk7RDggDuuAMuvzxsQPz005CZGXVVkpTeDHaStltFBdx0E/TtC/37w7BhULNm1FVJkmx3Imm7rFsHl10W7iRx331wzTVRVyRJ+oHBTtI2W70azjkHCgvhqafC7cKqs1gsRiwWo7y8POpSJCkhMoIgCKIuQlL1t2IFdO4M77wDY8eGTYiTRUlJCdnZ2RQXF5OVlRV1OZIUN47YSdqqr74KtwhbsiTcIqxNm6grkiRtisFO0hZ9/HE4OldeHm4R1qxZ1BVJkjbHVbGSNuutt8ItwurUgZkzDXWSVN0Z7CRt0ssvw3HHwUEHhVuENW4cdUWSpK0x2EnayOOPhwslTjghvKfuZz+LuiJJ0rYw2EmqFARQUACXXAKXXhqufq1bN+qqJEnbymAnCQgXR1x3HfzhD+FuEiNGwC4ur5KkpGKwk8TatWGz4WHD4IEHYMAAyMio+usEQcCAAQPYd999qVOnDsceeyzvv//+Fs956KGHaN++PXvuuSd77rknHTp04K233qr64iQpBRjspDT33Xdw4onwr3+FU69XXRW/aw0ZMoShQ4dy//33M3v2bBo0aEDHjh1ZuXLlZs+ZMmUK559/PpMnT+aNN95g//33Jz8/n6+++ip+hUpSknLnCSmN/dB4+Kuv4J//DFubxEsQBOy777706tWLW265BYDS0lJycnK48847uWobE2V5eTl77rkn999/P926ddumc9x5QlK6cMROSlPz50PbtlBcDK+/Ht9QB7Bw4UKKiorI/9FeZJmZmRxzzDHMnDlzm19n9erVrFu3jr322muzzyktLaWkpGSDhySlA4OdlIZefx2OPhr22APeeCMxjYeLiooAyMnJ2eB4Tk5O5e9tiz59+tCoUSM6dOiw2ecUFBSQnZ1d+WhsEz5JacJgJ6WZF1+EDh3gsMNg2jRo1Cg+1xk1ahS777575WPdunUAZPxkVUYQBBsd25whQ4bw9NNPM3bsWGrXrr3Z5/Xt25fi4uLKx+LFi3f8jUhSErGZgZRGHnwQevSAs86Cv/8dtpCNdtppp51GmzZtKn8uLS0FwpG7hg0bVh5funTpRqN4m3LXXXcxePBgXn31VQ477LAtPjczM5PMzMwdrFySkpcjdlIaCIKwN93VV0PPnvDMM/ENdQD16tXjoIMOqnzk5ubSoEEDCgsLK59TVlbG1KlTadeu3RZf6y9/+Qt/+tOfmDBhAq1bt45v4ZKUxByxk1Lc+vXQvTs8/DDccQf07h2fHnVbk5GRQa9evRg8eDBNmzaladOmDB48mLp169K1a9fK53Xr1o1GjRpRUFAAhNOvt912G0899RQHHHBA5f14P0zxSpL+x2AnpbDVq6FLF5gwIdz/dRu7g8RN7969WbNmDT169ODbb7+lTZs2TJw4kXr16lU+Z9GiRdSo8b/JhGHDhlFWVsZvf/vbDV6rf//+DBgwIFGlS1JSsI+dlKKWL4dTT4X33oPnnw/71aUr+9hJSheO2EkpaMEC6NQJvv0WJk+GvLyoK5IkJYKLJ6QU89ZbcOSR4YKJmTMNdZKUTgx2UgoZNw6OPRYOOigMdQcdFHVFkqREMthJKWLYMDjzzHAK9rXXoH79qCuSJCWawU5KchUVcMstYX+6666DZ5+FOnWirkqSFAUXT0hJbO1auOSSMMz99a/Qq1fUFUmSomSwk5LUihXh1Otbb8Fzz8HZZ0ddkSQpagY7KQl9/nl4L91//hPeT7eVHbnSViwWIxaLUV5eHnUpkpQQNiiWksycOXDKKbDbbvDyy3DwwVFXVP3ZoFhSunDxhJREXnoJfvMbaNIE3njDUCdJ2pDBTkoSI0bAaadBx47hbhL77BN1RZKk6sZgJ1VzFRXQpw9cdRV07w5jxkDdulFXJUmqjlw8IVVja9ZAt25hmLv7brjhBsjIiLoqSVJ1ZbCTqqlvvgmnXv/9bxg7Fs44I+qKJEnVncFOqobefz9c+VpaCtOmQatWUVckSUoG3mMnVTOFhWFfuqwsePNNQ50kadsZ7KRq5KGHwsbD7drBjBmw//5RVyRJSiYGO6kaqKiAW26BK68MH//8ZzhiJ0nS9vAeOyliq1eHK1/HjoWhQ6FXL1e+SpJ2jMFOilBREZx+erjy9YUXwl9LkrSjDHZSRFz5Gn+xWIxYLEZ5eXnUpUhSQmQEQRBEXYSUbgoL4be/Dfd8/de/XCQRbyUlJWRnZ1NcXEyWNy9KSmEunpASbMQIV75KkuLDYCclyPr14ZZgV10VPlz5Kkmqat5jJyVAcTGcd144BXv//dCzZ9QVSZJSkcFOirPPPoPOneHrr+Hll6Fjx6grkiSlKqdipTiaMgWOOCKchn3zTUOdJCm+DHZSnDz0UBjkDj8cZs2CX/4y6ookSanOYCdVsfXrw90jrrwSrrginH7da6+oq5IkpQPvsZOqUHExdOkCr77qIglJUuIZ7KQq8sMiiSVLXCQhSYqGU7FSFfjxIolZswx1kqRoGOyknfTjRRJvvukiCUlSdAx20g768SKJK68Mp1/33DPqqiRJ6cx77KQd8N13cP754U4SsRj06BF1RdqUWCxGLBajvLw86lIkKSEygiAIoi5CSiYffACnnw7/+Q889xx06BB1RdqakpISsrOzKS4uJssNeiWlMKdipe0wbhy0aQO1asHs2YY6SVL1YrCTtkFFBQwaFI7UdegAb7wBBx0UdVWSJG3Ie+ykrVi5Ei6+GF54IQx3/fpBDf9JJEmqhgx20hZ8+imccQYsWgT/+AecdlrUFUmStHmOO0ib8corkJcHZWVhfzpDnSSpujPYST8RBPCXv8DJJ0O7dvDWW9CsWdRVSZK0dQY76UdWr4YLLoDevaFPn3AV7B57RF2VJEnbxmAn/dcXX8DRR4f30j37LPz5z1CzZtRVpZYgCBgwYAD77rsvderU4dhjj+X999/f5vOfeeYZMjIyOOOMM+JYpSQlL4OdBEyZAq1bw7ffwsyZcM45UVeUmoYMGcLQoUO5//77mT17Ng0aNKBjx46sXLlyq+d+8cUX3HTTTbRv3z4BlUpScjLYKa0FAdx/f9ib7le/grffDv+rqhcEAffccw/9+vXjrLPOokWLFjz++OOsXr2ap556aovnlpeXc8EFFzBw4EB+/vOfJ6hiSUo+BjulrdWr4dJL4dpr4brrYMIE+NnPoq4qdS1cuJCioiLy8/Mrj2VmZnLMMccwc+bMLZ47aNAg9t57by677LJtulZpaSklJSUbPCQpHdjHTmlpwQI46yz4+GP4+9/hwgujrij1FRUVAZCTk7PB8ZycHL744ovNnvf666/zyCOPMG/evG2+VkFBAQMHDtyxQiUpiTlip7Qzfjy0agXffw+zZhnq4mXUqFHsvvvulY9169YBkJGRscHzgiDY6NgPVq5cyYUXXshDDz1E/fr1t/naffv2pbi4uPKxePHiHX8jkpREHLFT2vhhv9eBA6FzZ3jiCVuZxNNpp51GmzZtKn8uLS0FwpG7hg0bVh5funTpRqN4P/jss8/4/PPP6dy5c+WxiooKAHbZZRc++ugjfvGLX2x0XmZmJpmZmVXyPiQpmRjslBZWrAhH5iZMgNtvh7593e813urVq0e9evUqfw6CgAYNGlBYWMjhhx8OQFlZGVOnTuXOO+/c5GsccsghvPfeexscu/XWW1m5ciX33nsvjRs3jt8bkKQkZLBTyps7N7yfrqQkDHY/undfCZSRkUGvXr0YPHgwTZs2pWnTpgwePJi6devStWvXyud169aNRo0aUVBQQO3atWnRosUGr7PHf4dZf3pckmSwU4obORK6d4fmzWHyZDjggKgrSm+9e/dmzZo19OjRg2+//ZY2bdowceLEDUb2Fi1aRA2HUyVph2QEQRBEXYRU1UpL4frr4cEH4bLLwl51tWtHXZWiUlJSQnZ2NsXFxWRlZUVdjiTFjSN2SjmLF8PZZ8M778BDD8Hll0ddkSRJiWGwU0qZOBEuuADq1oUZMyAvL+qKJElKHG9kUUooL4c//hFOOinsUTdnjqFOkpR+HLFT0isqgq5dYepU+NOfbGUiSUpfBjsltcmTw1AH8NprcOyxkZYjSVKkHNdQUqqogD//GTp0gGbNwl51hjpJUroz2CnpLFsGJ58Mt90G/fpBYSE0aBB1VZIkRc+pWCWV11+HLl3CPnXuIqGticVixGIxysvLoy5FkhLCBsVKCkEAd98NffpA27bwzDPQqFHUVSlZ2KBYUrpwKlbV3ooVcPrpcPPN8Pvfw6RJhjpJkjbFqVhVa2+9BeeeCyUl8M9/wqmnRl2RJEnVlyN2qpYqKsKp16OPhpyccNWroU6SpC0z2KnaWboUTjkFbroJrr8epk+HJk2irkqSpOrPqVhVK6++ChddFI7YvfxyuEWYJEnaNo7YqVpYty7cCiw/H1q0gHfeMdRJkrS9HLFT5BYuhPPPhzlzoKAgXP3qXq+SJG0/g50iNXo0XHkl7LUXzJgBbdpEXZEkScnLcRFFYtUquOIKOO886NQJ5s0z1EmStLMcsVPCvftuuC3YF1/Aww/D734HGRlRVyVJUvJzxE4JEwQQi8ERR0CtWuE9dZddZqiTJKmqGOyUEMuWwVlnwTXXwOWXw5tvQrNmUVelVBeLxcjNzSUvLy/qUiQpITKCIAiiLkKpbeJEuOQSKC2FRx6BM86IuiKlm5KSErKzsykuLiYrKyvqciQpbhyxU9ysWRPuHHHiiXDoofDee4Y6SZLiycUTiot33oELLoBPP4V77w2nYO1NJ0lSfPlXrapURQXcfXe4QKJmTXj7bbjuOkOdJEmJ4F+3qjJffgkdO8JNN8G118Jbb4Xbg0mSpMRwKlZV4rnn4KqroG5dePVVOOGEqCuSJCn9OGKnnVJSAhdfDOeeG4a5d9811EmSFBVH7LTDJk8O25isWAEjR0K3bjYbliQpSo7YabutXh22MTn+eDjwwHCU7uKLDXWSJEXNETttlzffDEfmvvgChg4NA54rXiVJqh78K1nbpKwM+vWDdu0gOxvmzoUbbjDUSZJUnThip616991wlO7992HAAOjbF3bxkyNJUrXjeIs2a/16uOMOaN06/PWbb8JttxnqJEmqrgx22qRPPoH27eEPfwinXOfMgZYto65K2j6xWIzc3Fzy8vKiLkWSEiIjCIIg6iJUfVRUwLBh0Ls37LsvPPFEeF+dlMxKSkrIzs6muLiYrKysqMuRpLhxxE6VPv00bGFy7bVw6aXwzjuGOkmSkonBTpSXw1//CocdFrYxefVViMVgt92irkySJG0Pg12a+/DD8F663/8errgC3nvPLcEkSUpWBrs09cOK11//GpYvh2nT4N57Yffdo65MkiTtKINdGnr3XTjyyLDh8HXXwbx5cPTRUVclSZJ2lsEujZSVhQ2GW7WCtWvhjTdgyBCoUyfqyiRJUlWw1WyamDMnXOn6wQfhzhH9+kFmZtRVSZKkquSIXYpbuzYMcm3ahDtGzJ4NgwYZ6hSNIAgYMGAA++67L3Xq1OHYY4/l/fff3+p53333HT179qRhw4bUrl2bZs2aMX78+ARULEnJxRG7FDZtGlx5JSxcGIa5m2+GXXeNuiqlsyFDhjB06FBGjhzJwQcfzO23307Hjh356KOPqFev3ibPKSsro2PHjuyzzz48//zz7LfffixevHizz5ekdGawS0HffhvuHPHww9C2LYwZA82bR12V0l0QBNxzzz3069ePs846C4DHH3+cnJwcnnrqKa666qpNnvfoo4+yYsUKZs6cya7//ZdJkyZNEla3JCUTp2JTSBDA6NHQrBk8+2y4NdiMGYY6VQ8LFy6kqKiI/Pz8ymOZmZkcc8wxzJw5c7PnjRs3jrZt29KzZ09ycnJo0aIFgwcPpry8fLPnlJaWUlJSssFDktKBwS5FfPEFnHoqnHde2Lrkgw+ge3eo4f/CqiaKiooAyMnJ2eB4Tk5O5e9tyoIFC3j++ecpLy9n/Pjx3Hrrrdx99938+c9/3uw5BQUFZGdnVz4aN25cNW9Ckqo5/9pPcuvXh9uB5eaGe7u++CI8/zzsu2/UlSndjRo1it13373ysW7dOgAyMjI2eF4QBBsd+7GKigr22WcfRowYQatWrTjvvPPo168fw4cP3+w5ffv2pbi4uPKxePHiqnlTklTNeY9dEps7N9wG7P/+D665Bm6/HbKyoq5KCp122mm0adOm8ufS0lIgHLlr2LBh5fGlS5duNIr3Yw0bNmTXXXelZs2alceaNWtGUVERZWVl1KpVa6NzMjMzyXTpt6Q05IhdElq1KlzhmpcXNh1+4w34298Mdape6tWrx0EHHVT5yM3NpUGDBhQWFlY+p6ysjKlTp9KuXbvNvs5RRx3Fp59+SkVFReWxjz/+mIYNG24y1ElSOjPYJZl//QtatID77w9H6ObMCXvUSdVdRkYGvXr1YvDgwbzwwgv8+9//5pJLLqFu3bp07dq18nndunWjb9++lT93796d5cuXc/311/Pxxx/z0ksvMXjwYHr27BnF25Ckas2p2CTxxRdw/fXwj39Ax45QWAgHHRR1VdL26d27N2vWrKFHjx58++23tGnThokTJ27Qk27RokXU+NGqn8aNGzNx4kRuuOEGDjvsMBo1asT111/PLbfcEsVbkKRqLSMIgiDqIrR5ZWUwdGjYYHivvcKFEr/9LWzhXnNJP1FSUkJ2djbFxcVkec+CpBTmiF01NmkS9OwJn3wCvXpB//5gs31JkrQ53mNXDS1ZAhdcACecAD/7Wbj69a67DHWSJGnLDHbVyPr14erWQw6BiRPhscfC/V4PPTTqyiRJUjIw2FUTs2aF7Ut69YKuXeGjj+CSS9w5QpIkbTtjQ8SWL4crr4S2baFmTXjzTRg+PFwoIUmStD1cPBGR9ethxAi49VaoqIBYDK66Kgx3kiRJO8IRuwhMmQItW4bbgJ19Nnz8MfToYaiTJEk7x2CXQF98AeeeC8cdB7vvDm+9BQ89BPvsE3VlUmqKxWLk5uaSl5cXdSmSlBA2KE6ANWtgyBC44w7Yc0+4886wnYkLI6TEsEGxpHThPXZxFAQwZgzcdBN8/TXceCP062c/OkmSFB8Guzh5771wb9fJk+HUU8O9XZs2jboqSZKUypwMrGIrVsB118Hhh8OXX8JLL8E//2mokyRJ8eeIXRUpKwv7zw0cCOvWQUFBOGJXq1bUlUmSpHRhsNtJQQD/+AfcfDMsWACXXQaDBkGDBlFXJkmS0o1TsTvh//4vbF1y5plw4IEwb17YdNhQJ0mSomCw2wFffgkXXwytW8N//gPjx8Mrr8Chh0ZdmSRJSmdOxW6H77+Hv/wlfOy+OwwbBpdfDrv4pyhJkqoBI8k2KC+Hxx8P93VdsQJuuAH69gX7nEqSpOrEqditeO01aNUqXBRx7LHw4YfhildDnSRJqm4Mdpsxdy6ceCJ06AC77QazZsFTT8EBB0RdmSRJ0qYZ7H5iwQLo2hVatoQvvoCxY2HGDGjTJurKJG2vWCxGbm4ueXl5UZciSQmREQRBEHUR1cHSpXD77fDAA7D33mGj4UsucWGElApKSkrIzs6muLiYLO+jkJTC0j62rFwJQ4fCXXdBzZphc+HrroO6daOuTJIkafukbbArKwubCf/pT1BcDNdeG6503WuvqCuTJEnaMWl3j11FBTzzDDRrFu7lesop8PHHYW86Q50kSUpmaRPsgiDcHaJ1azj/fGjeHN55Bx59FPbfP+rqJEmSdl5aBLupU+E3v4GTToI6dWD6dBg3Dlq0iLoySZKkqpPSwW7WLOjYMWwsvHp1uKfrjBlw9NFRVyZJklT1UjLYzZ0LnTtD27awZEnYi+7tt6FTJ8jIiLo6SZKk+EipYDd/PpxzTthc+KOPwp0i3nkHzjzTQCdJklJfSgS7Tz+Fiy4K75mbPTtcEDF/frhIombNqKuTJElKjKQOdl98AVdcAYccApMmQSwWti659FJ3jJAkSenn/9u7v9AmrzCO499MWNZqbEEkpZIboRbEDXohlVFRBu2N1ILCEC+6waBSp1BRtLKKQteWVigytOxGNqfdKAy9mDimEyoyGRULIrVuqEVBqHghNmKNlb27CIQ5zf65NMnb7wdy8R5ykid3v7znnPcpyvgzMQHd3fDll7LGv0oAAARzSURBVFBeDn190NqaPvEqSZI0VxVVsLt1C7q64KuvYNEi6OlJB7r58/NdmSRJUv4VxVLsr7/CBx9AdTV8/326S8TEBOzaZaiTlN2RI0dYvnw5K1euzHcpkjQrIkEQBPkuIpsbN+DTT+GbbyAeh/b29J46l1wl/RtTU1OUlZXx6NEjFi5cmO9yJClnCnIp9vp16OyEoSGorITPPoOPPoK33sp3ZZIkSYWroJZir12D999PP7bkp5/Sp1xv3YKPPzbUSZIk/Z2CCHY//wxNTfDOOzAyAp9/nn42XWsrRKP5rk6SJKk45C3YBQH8+CO891669dcvv8AXX6QPSrS0wJtv5qsySZKk4jTrwe633+DUKaithfp6mJqCb7+FsTH48EMDnSRJ0n81a8FuZgaOH4e334YNG6C0FH74Id0CbONGW39JkiS9rpwHu+lpGBiAZcuguRmWLk0fjBgehoYGiERyXYEkSdLckPNg19IC27en99FdvQrffQfvvpvrb5VUiIIg4MCBA1RWVlJSUsLatWsZGxv723mHDh2iurqakpISEokEO3bs4OnTp7NQsSQVl5wHu08+SR+M+Prr9KlXSXNXX18f/f39HD58mMuXL1NRUUF9fT3JZDLrnMHBQdrb29m/fz/j4+McPXqUoaEh9u7dO4uVS1JxKOjOE5LCIwgCKisraWtrY8+ePQCkUini8Ti9vb1s2bLllfO2bdvG+Pg458+fz4zt3LmTkZERLl68+I++284TkuaKgniOnaTwm5iYYHJykoaGhsxYNBplzZo1XLp0Keu8uro6rly5wsjICAC3b9/mzJkzrFu3Luc1S1KxKciWYpLCZ3JyEoB4PP7CeDwe586dO1nnbdq0iQcPHlBXV0cQBDx//pzW1lba29uzzkmlUqRSqcz11NTUa1YvScXBO3aScmJwcJAFCxZkXjMzMwBE/nQUPgiCl8b+aHh4mK6uLgYGBhgdHeXkyZOcPn2azs7OrHN6enooKyvLvBKJxP/zoySpwLnHTlJOJJNJ7t+/n7lOpVKsWLGC0dFRampqMuNNTU2Ul5dz7NixV37O6tWrWbVqFQcPHsyMnThxgpaWFh4/fswbb7z8//RVd+wSiYR77CSFnkuxknIiFosRi8Uy10EQUFFRwblz5zLB7tmzZ1y4cIHe3t6sn/PkyZOXwtu8efMIgoBs/0uj0ShRG01LmoMMdpJmRSQSoa2tje7ubqqqqqiqqqK7u5vS0lI2b96ceV9zczNLliyhp6cHgMbGRvr7+6mpqaG2tpabN2+yb98+1q9fzzxb1kjSCwx2kmbN7t27mZ6eZuvWrTx8+JDa2lrOnj37wp29u3fvvnCHrqOjg0gkQkdHB/fu3WPx4sU0NjbS1dWVj58gSQXNPXaSQi8IApLJJLFY7C8PakhSsTPYSZIkhYSPO5EkSQoJg50kSVJIGOwkSZJCwmAnSZIUEgY7SZKkkDDYSZIkhYTBTpIkKSQMdpIkSSFhsJMkSQoJg50kSVJIGOwkSZJC4ndV8x3DgFS/5AAAAABJRU5ErkJggg==\n",
      "text/plain": [
       "Graphics object consisting of 1 graphics primitive"
      ]
     },
     "execution_count": 47,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "plot(arctan(x),)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 48,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": 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\n",
      "text/plain": [
       "Graphics object consisting of 1 graphics primitive"
      ]
     },
     "execution_count": 48,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "plot(erf(x),-10,10)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 49,
   "metadata": {},
   "outputs": [
    {
     "ename": "TypeError",
     "evalue": "cannot coerce arguments: no canonical coercion from <class 'NoneType'> to Symbolic Ring",
     "output_type": "error",
     "traceback": [
      "\u001b[0;31m---------------------------------------------------------------------------\u001b[0m",
      "\u001b[0;31mTypeError\u001b[0m                                 Traceback (most recent call last)",
      "\u001b[0;32m/home/abhigyan/anaconda3/envs/sage/lib/python3.7/site-packages/sage/symbolic/function.pyx\u001b[0m in \u001b[0;36msage.symbolic.function.Function.__call__ (build/cythonized/sage/symbolic/function.cpp:6517)\u001b[0;34m()\u001b[0m\n\u001b[1;32m    461\u001b[0m             \u001b[0;32mtry\u001b[0m\u001b[0;34m:\u001b[0m\u001b[0;34m\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[0;32m--> 462\u001b[0;31m                 \u001b[0margs\u001b[0m \u001b[0;34m=\u001b[0m \u001b[0;34m[\u001b[0m\u001b[0mSR\u001b[0m\u001b[0;34m.\u001b[0m\u001b[0mcoerce\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0ma\u001b[0m\u001b[0;34m)\u001b[0m \u001b[0;32mfor\u001b[0m \u001b[0ma\u001b[0m \u001b[0;32min\u001b[0m \u001b[0margs\u001b[0m\u001b[0;34m]\u001b[0m\u001b[0;34m\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[0m\u001b[1;32m    463\u001b[0m             \u001b[0;32mexcept\u001b[0m \u001b[0mTypeError\u001b[0m \u001b[0;32mas\u001b[0m \u001b[0merr\u001b[0m\u001b[0;34m:\u001b[0m\u001b[0;34m\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n",
      "\u001b[0;32m/home/abhigyan/anaconda3/envs/sage/lib/python3.7/site-packages/sage/structure/parent.pyx\u001b[0m in \u001b[0;36msage.structure.parent.Parent.coerce (build/cythonized/sage/structure/parent.c:10550)\u001b[0;34m()\u001b[0m\n\u001b[1;32m   1113\u001b[0m \u001b[0;34m\u001b[0m\u001b[0m\n\u001b[0;32m-> 1114\u001b[0;31m     \u001b[0mcpdef\u001b[0m \u001b[0mcoerce\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0mself\u001b[0m\u001b[0;34m,\u001b[0m \u001b[0mx\u001b[0m\u001b[0;34m)\u001b[0m\u001b[0;34m:\u001b[0m\u001b[0;34m\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[0m\u001b[1;32m   1115\u001b[0m         \"\"\"\n",
      "\u001b[0;32m/home/abhigyan/anaconda3/envs/sage/lib/python3.7/site-packages/sage/structure/parent.pyx\u001b[0m in \u001b[0;36msage.structure.parent.Parent.coerce (build/cythonized/sage/structure/parent.c:10479)\u001b[0;34m()\u001b[0m\n\u001b[1;32m   1143\u001b[0m                     \u001b[0m_record_exception\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0;34m)\u001b[0m\u001b[0;34m\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[0;32m-> 1144\u001b[0;31m             \u001b[0;32mraise\u001b[0m \u001b[0mTypeError\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0m_LazyString\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0m_lazy_format\u001b[0m\u001b[0;34m,\u001b[0m \u001b[0;34m(\u001b[0m\u001b[0;34m\"no canonical coercion from %s to %s\"\u001b[0m\u001b[0;34m,\u001b[0m \u001b[0mparent\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0mx\u001b[0m\u001b[0;34m)\u001b[0m\u001b[0;34m,\u001b[0m \u001b[0mself\u001b[0m\u001b[0;34m)\u001b[0m\u001b[0;34m,\u001b[0m \u001b[0;34m{\u001b[0m\u001b[0;34m}\u001b[0m\u001b[0;34m)\u001b[0m\u001b[0;34m)\u001b[0m\u001b[0;34m\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[0m\u001b[1;32m   1145\u001b[0m         \u001b[0;32melse\u001b[0m\u001b[0;34m:\u001b[0m\u001b[0;34m\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n",
      "\u001b[0;31mTypeError\u001b[0m: no canonical coercion from <class 'NoneType'> to Symbolic Ring",
      "\nDuring handling of the above exception, another exception occurred:\n",
      "\u001b[0;31mTypeError\u001b[0m                                 Traceback (most recent call last)",
      "\u001b[0;32m<ipython-input-49-17eb686cc068>\u001b[0m in \u001b[0;36m<module>\u001b[0;34m()\u001b[0m\n\u001b[0;32m----> 1\u001b[0;31m \u001b[0mintegral\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0mx\u001b[0m\u001b[0;34m*\u001b[0m\u001b[0me\u001b[0m\u001b[0;34m**\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0mInteger\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0;36m2\u001b[0m\u001b[0;34m)\u001b[0m\u001b[0;34m*\u001b[0m\u001b[0my\u001b[0m\u001b[0;34m)\u001b[0m\u001b[0;34m/\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0mInteger\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0;36m4\u001b[0m\u001b[0;34m)\u001b[0m\u001b[0;34m-\u001b[0m\u001b[0my\u001b[0m\u001b[0;34m)\u001b[0m\u001b[0;34m,\u001b[0m\u001b[0;34m)\u001b[0m\u001b[0;34m\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[0m",
      "\u001b[0;32m/home/abhigyan/anaconda3/envs/sage/lib/python3.7/site-packages/sage/misc/functional.py\u001b[0m in \u001b[0;36mintegral\u001b[0;34m(x, *args, **kwds)\u001b[0m\n\u001b[1;32m    751\u001b[0m     \"\"\"\n\u001b[1;32m    752\u001b[0m     \u001b[0;32mif\u001b[0m \u001b[0mhasattr\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0mx\u001b[0m\u001b[0;34m,\u001b[0m \u001b[0;34m'integral'\u001b[0m\u001b[0;34m)\u001b[0m\u001b[0;34m:\u001b[0m\u001b[0;34m\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[0;32m--> 753\u001b[0;31m         \u001b[0;32mreturn\u001b[0m \u001b[0mx\u001b[0m\u001b[0;34m.\u001b[0m\u001b[0mintegral\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0;34m*\u001b[0m\u001b[0margs\u001b[0m\u001b[0;34m,\u001b[0m \u001b[0;34m**\u001b[0m\u001b[0mkwds\u001b[0m\u001b[0;34m)\u001b[0m\u001b[0;34m\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[0m\u001b[1;32m    754\u001b[0m     \u001b[0;32melse\u001b[0m\u001b[0;34m:\u001b[0m\u001b[0;34m\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[1;32m    755\u001b[0m         \u001b[0;32mfrom\u001b[0m \u001b[0msage\u001b[0m\u001b[0;34m.\u001b[0m\u001b[0msymbolic\u001b[0m\u001b[0;34m.\u001b[0m\u001b[0mring\u001b[0m \u001b[0;32mimport\u001b[0m \u001b[0mSR\u001b[0m\u001b[0;34m\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n",
      "\u001b[0;32m/home/abhigyan/anaconda3/envs/sage/lib/python3.7/site-packages/sage/symbolic/expression.pyx\u001b[0m in \u001b[0;36msage.symbolic.expression.Expression.integral (build/cythonized/sage/symbolic/expression.cpp:64072)\u001b[0;34m()\u001b[0m\n\u001b[1;32m  12404\u001b[0m                     \u001b[0mR\u001b[0m \u001b[0;34m=\u001b[0m \u001b[0mring\u001b[0m\u001b[0;34m.\u001b[0m\u001b[0mSR\u001b[0m\u001b[0;34m\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[1;32m  12405\u001b[0m             \u001b[0;32mreturn\u001b[0m \u001b[0mR\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0mintegral\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0mf\u001b[0m\u001b[0;34m,\u001b[0m \u001b[0mv\u001b[0m\u001b[0;34m,\u001b[0m \u001b[0ma\u001b[0m\u001b[0;34m,\u001b[0m \u001b[0mb\u001b[0m\u001b[0;34m,\u001b[0m \u001b[0;34m**\u001b[0m\u001b[0mkwds\u001b[0m\u001b[0;34m)\u001b[0m\u001b[0;34m)\u001b[0m\u001b[0;34m\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[0;32m> 12406\u001b[0;31m         \u001b[0;32mreturn\u001b[0m \u001b[0mintegral\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0mself\u001b[0m\u001b[0;34m,\u001b[0m \u001b[0;34m*\u001b[0m\u001b[0margs\u001b[0m\u001b[0;34m,\u001b[0m \u001b[0;34m**\u001b[0m\u001b[0mkwds\u001b[0m\u001b[0;34m)\u001b[0m\u001b[0;34m\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[0m\u001b[1;32m  12407\u001b[0m \u001b[0;34m\u001b[0m\u001b[0m\n\u001b[1;32m  12408\u001b[0m     \u001b[0mintegrate\u001b[0m \u001b[0;34m=\u001b[0m \u001b[0mintegral\u001b[0m\u001b[0;34m\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n",
      "\u001b[0;32m/home/abhigyan/anaconda3/envs/sage/lib/python3.7/site-packages/sage/symbolic/integration/integral.py\u001b[0m in \u001b[0;36mintegrate\u001b[0;34m(expression, v, a, b, algorithm, hold)\u001b[0m\n\u001b[1;32m    824\u001b[0m         \u001b[0;32mreturn\u001b[0m \u001b[0mintegrator\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0mexpression\u001b[0m\u001b[0;34m,\u001b[0m \u001b[0mv\u001b[0m\u001b[0;34m,\u001b[0m \u001b[0ma\u001b[0m\u001b[0;34m,\u001b[0m \u001b[0mb\u001b[0m\u001b[0;34m)\u001b[0m\u001b[0;34m\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[1;32m    825\u001b[0m     \u001b[0;32mif\u001b[0m \u001b[0ma\u001b[0m \u001b[0;32mis\u001b[0m \u001b[0;32mNone\u001b[0m\u001b[0;34m:\u001b[0m\u001b[0;34m\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[0;32m--> 826\u001b[0;31m         \u001b[0;32mreturn\u001b[0m \u001b[0mindefinite_integral\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0mexpression\u001b[0m\u001b[0;34m,\u001b[0m \u001b[0mv\u001b[0m\u001b[0;34m,\u001b[0m \u001b[0mhold\u001b[0m\u001b[0;34m=\u001b[0m\u001b[0mhold\u001b[0m\u001b[0;34m)\u001b[0m\u001b[0;34m\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[0m\u001b[1;32m    827\u001b[0m     \u001b[0;32melse\u001b[0m\u001b[0;34m:\u001b[0m\u001b[0;34m\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[1;32m    828\u001b[0m         \u001b[0;32mreturn\u001b[0m \u001b[0mdefinite_integral\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0mexpression\u001b[0m\u001b[0;34m,\u001b[0m \u001b[0mv\u001b[0m\u001b[0;34m,\u001b[0m \u001b[0ma\u001b[0m\u001b[0;34m,\u001b[0m \u001b[0mb\u001b[0m\u001b[0;34m,\u001b[0m \u001b[0mhold\u001b[0m\u001b[0;34m=\u001b[0m\u001b[0mhold\u001b[0m\u001b[0;34m)\u001b[0m\u001b[0;34m\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n",
      "\u001b[0;32m/home/abhigyan/anaconda3/envs/sage/lib/python3.7/site-packages/sage/symbolic/function.pyx\u001b[0m in \u001b[0;36msage.symbolic.function.BuiltinFunction.__call__ (build/cythonized/sage/symbolic/function.cpp:11848)\u001b[0;34m()\u001b[0m\n\u001b[1;32m    996\u001b[0m             \u001b[0mres\u001b[0m \u001b[0;34m=\u001b[0m \u001b[0mself\u001b[0m\u001b[0;34m.\u001b[0m\u001b[0m_evalf_try_\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0;34m*\u001b[0m\u001b[0margs\u001b[0m\u001b[0;34m)\u001b[0m\u001b[0;34m\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[1;32m    997\u001b[0m             \u001b[0;32mif\u001b[0m \u001b[0mres\u001b[0m \u001b[0;32mis\u001b[0m \u001b[0;32mNone\u001b[0m\u001b[0;34m:\u001b[0m\u001b[0;34m\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[0;32m--> 998\u001b[0;31m                 res = super(BuiltinFunction, self).__call__(\n\u001b[0m\u001b[1;32m    999\u001b[0m                         *args, coerce=coerce, hold=hold)\n\u001b[1;32m   1000\u001b[0m \u001b[0;34m\u001b[0m\u001b[0m\n",
      "\u001b[0;32m/home/abhigyan/anaconda3/envs/sage/lib/python3.7/site-packages/sage/symbolic/function.pyx\u001b[0m in \u001b[0;36msage.symbolic.function.Function.__call__ (build/cythonized/sage/symbolic/function.cpp:6671)\u001b[0;34m()\u001b[0m\n\u001b[1;32m    473\u001b[0m                     \u001b[0;32mif\u001b[0m \u001b[0mcallable\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0mmethod\u001b[0m\u001b[0;34m)\u001b[0m\u001b[0;34m:\u001b[0m\u001b[0;34m\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[1;32m    474\u001b[0m                         \u001b[0;32mreturn\u001b[0m \u001b[0mmethod\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0;34m)\u001b[0m\u001b[0;34m\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[0;32m--> 475\u001b[0;31m                 \u001b[0;32mraise\u001b[0m \u001b[0mTypeError\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0;34m\"cannot coerce arguments: %s\"\u001b[0m \u001b[0;34m%\u001b[0m \u001b[0;34m(\u001b[0m\u001b[0merr\u001b[0m\u001b[0;34m)\u001b[0m\u001b[0;34m)\u001b[0m\u001b[0;34m\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[0m\u001b[1;32m    476\u001b[0m \u001b[0;34m\u001b[0m\u001b[0m\n\u001b[1;32m    477\u001b[0m         \u001b[0;32melse\u001b[0m\u001b[0;34m:\u001b[0m \u001b[0;31m# coerce == False\u001b[0m\u001b[0;34m\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n",
      "\u001b[0;31mTypeError\u001b[0m: cannot coerce arguments: no canonical coercion from <class 'NoneType'> to Symbolic Ring"
     ]
    }
   ],
   "source": [
    "integral(x*e^(2*y)/(4-y),)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 53,
   "metadata": {},
   "outputs": [],
   "source": [
    "[x11,x22,x33]= solve(a*x^3+b*x^2+c*x+d==0,x)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 54,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/html": [
       "<html><script type=\"math/tex; mode=display\">\\newcommand{\\Bold}[1]{\\mathbf{#1}}x = -\\frac{{\\left(-i \\, \\sqrt{3} + 1\\right)} {\\left(\\frac{b^{2}}{a^{2}} - \\frac{3 \\, c}{a}\\right)}}{18 \\, {\\left(-\\frac{b^{3}}{27 \\, a^{3}} + \\frac{b c}{6 \\, a^{2}} - \\frac{d}{2 \\, a} + \\frac{\\sqrt{-\\frac{1}{3} \\, b^{2} c^{2} + \\frac{4}{3} \\, a c^{3} + 9 \\, a^{2} d^{2} + \\frac{2}{3} \\, {\\left(2 \\, b^{3} - 9 \\, a b c\\right)} d}}{6 \\, a^{2}}\\right)}^{\\frac{1}{3}}} - \\frac{1}{2} \\, {\\left(i \\, \\sqrt{3} + 1\\right)} {\\left(-\\frac{b^{3}}{27 \\, a^{3}} + \\frac{b c}{6 \\, a^{2}} - \\frac{d}{2 \\, a} + \\frac{\\sqrt{-\\frac{1}{3} \\, b^{2} c^{2} + \\frac{4}{3} \\, a c^{3} + 9 \\, a^{2} d^{2} + \\frac{2}{3} \\, {\\left(2 \\, b^{3} - 9 \\, a b c\\right)} d}}{6 \\, a^{2}}\\right)}^{\\frac{1}{3}} - \\frac{b}{3 \\, a}</script></html>"
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      "text/plain": [
       "x == -1/18*(-I*sqrt(3) + 1)*(b^2/a^2 - 3*c/a)/(-1/27*b^3/a^3 + 1/6*b*c/a^2 - 1/2*d/a + 1/6*sqrt(-1/3*b^2*c^2 + 4/3*a*c^3 + 9*a^2*d^2 + 2/3*(2*b^3 - 9*a*b*c)*d)/a^2)^(1/3) - 1/2*(I*sqrt(3) + 1)*(-1/27*b^3/a^3 + 1/6*b*c/a^2 - 1/2*d/a + 1/6*sqrt(-1/3*b^2*c^2 + 4/3*a*c^3 + 9*a^2*d^2 + 2/3*(2*b^3 - 9*a*b*c)*d)/a^2)^(1/3) - 1/3*b/a"
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     "metadata": {},
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   ],
   "source": [
    "x11.show()"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 55,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/html": [
       "<html><script type=\"math/tex; mode=display\">\\newcommand{\\Bold}[1]{\\mathbf{#1}}x = -\\frac{{\\left(i \\, \\sqrt{3} + 1\\right)} {\\left(\\frac{b^{2}}{a^{2}} - \\frac{3 \\, c}{a}\\right)}}{18 \\, {\\left(-\\frac{b^{3}}{27 \\, a^{3}} + \\frac{b c}{6 \\, a^{2}} - \\frac{d}{2 \\, a} + \\frac{\\sqrt{-\\frac{1}{3} \\, b^{2} c^{2} + \\frac{4}{3} \\, a c^{3} + 9 \\, a^{2} d^{2} + \\frac{2}{3} \\, {\\left(2 \\, b^{3} - 9 \\, a b c\\right)} d}}{6 \\, a^{2}}\\right)}^{\\frac{1}{3}}} - \\frac{1}{2} \\, {\\left(-i \\, \\sqrt{3} + 1\\right)} {\\left(-\\frac{b^{3}}{27 \\, a^{3}} + \\frac{b c}{6 \\, a^{2}} - \\frac{d}{2 \\, a} + \\frac{\\sqrt{-\\frac{1}{3} \\, b^{2} c^{2} + \\frac{4}{3} \\, a c^{3} + 9 \\, a^{2} d^{2} + \\frac{2}{3} \\, {\\left(2 \\, b^{3} - 9 \\, a b c\\right)} d}}{6 \\, a^{2}}\\right)}^{\\frac{1}{3}} - \\frac{b}{3 \\, a}</script></html>"
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      "text/plain": [
       "x == -1/18*(I*sqrt(3) + 1)*(b^2/a^2 - 3*c/a)/(-1/27*b^3/a^3 + 1/6*b*c/a^2 - 1/2*d/a + 1/6*sqrt(-1/3*b^2*c^2 + 4/3*a*c^3 + 9*a^2*d^2 + 2/3*(2*b^3 - 9*a*b*c)*d)/a^2)^(1/3) - 1/2*(-I*sqrt(3) + 1)*(-1/27*b^3/a^3 + 1/6*b*c/a^2 - 1/2*d/a + 1/6*sqrt(-1/3*b^2*c^2 + 4/3*a*c^3 + 9*a^2*d^2 + 2/3*(2*b^3 - 9*a*b*c)*d)/a^2)^(1/3) - 1/3*b/a"
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   ],
   "source": [
    "x22.show()"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 56,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/html": [
       "<html><script type=\"math/tex; mode=display\">\\newcommand{\\Bold}[1]{\\mathbf{#1}}x = \\frac{\\frac{b^{2}}{a^{2}} - \\frac{3 \\, c}{a}}{9 \\, {\\left(-\\frac{b^{3}}{27 \\, a^{3}} + \\frac{b c}{6 \\, a^{2}} - \\frac{d}{2 \\, a} + \\frac{\\sqrt{-\\frac{1}{3} \\, b^{2} c^{2} + \\frac{4}{3} \\, a c^{3} + 9 \\, a^{2} d^{2} + \\frac{2}{3} \\, {\\left(2 \\, b^{3} - 9 \\, a b c\\right)} d}}{6 \\, a^{2}}\\right)}^{\\frac{1}{3}}} + {\\left(-\\frac{b^{3}}{27 \\, a^{3}} + \\frac{b c}{6 \\, a^{2}} - \\frac{d}{2 \\, a} + \\frac{\\sqrt{-\\frac{1}{3} \\, b^{2} c^{2} + \\frac{4}{3} \\, a c^{3} + 9 \\, a^{2} d^{2} + \\frac{2}{3} \\, {\\left(2 \\, b^{3} - 9 \\, a b c\\right)} d}}{6 \\, a^{2}}\\right)}^{\\frac{1}{3}} - \\frac{b}{3 \\, a}</script></html>"
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      "text/plain": [
       "x == 1/9*(b^2/a^2 - 3*c/a)/(-1/27*b^3/a^3 + 1/6*b*c/a^2 - 1/2*d/a + 1/6*sqrt(-1/3*b^2*c^2 + 4/3*a*c^3 + 9*a^2*d^2 + 2/3*(2*b^3 - 9*a*b*c)*d)/a^2)^(1/3) + (-1/27*b^3/a^3 + 1/6*b*c/a^2 - 1/2*d/a + 1/6*sqrt(-1/3*b^2*c^2 + 4/3*a*c^3 + 9*a^2*d^2 + 2/3*(2*b^3 - 9*a*b*c)*d)/a^2)^(1/3) - 1/3*b/a"
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     "metadata": {},
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   ],
   "source": [
    "x33.show()"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": []
  }
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