{
 "cells": [
  {
   "cell_type": "code",
   "execution_count": 1,
   "id": "blind-lesbian",
   "metadata": {},
   "outputs": [],
   "source": [
    "# This short notebook will show how to take symbolic integrals in Python.\n",
    "# We will require the 'SymPy' module.\n",
    "import sympy as sym"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 2,
   "id": "opponent-concord",
   "metadata": {},
   "outputs": [],
   "source": [
    "# First, let's define a symbol x.\n",
    "x = sym.Symbol('x')"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 3,
   "id": "tropical-senegal",
   "metadata": {},
   "outputs": [],
   "source": [
    "# Then we can define f in terms of x.\n",
    "f = 2*x**2+3"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 4,
   "id": "formal-holder",
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "2(2)^2 + 3 = 11\n"
     ]
    }
   ],
   "source": [
    "# To evaluate f at a particular value of x, we can use 'subs()'.\n",
    "z = f.subs(x, 2)\n",
    "print('2(2)^2 + 3 =', z)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 5,
   "id": "outer-release",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/latex": [
       "$\\displaystyle \\frac{2 x^{3}}{3} + 3 x$"
      ],
      "text/plain": [
       "2*x**3/3 + 3*x"
      ]
     },
     "execution_count": 5,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "# To integrate f with respect to x, we use 'integrate()'.\n",
    "intf = f.integrate(x)\n",
    "intf"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 6,
   "id": "coordinated-distribution",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/latex": [
       "$\\displaystyle \\frac{2 x^{3}}{3} + 3 x$"
      ],
      "text/plain": [
       "2*x**3/3 + 3*x"
      ]
     },
     "execution_count": 6,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "# We can get the same result using 'integrate()' in the following way:\n",
    "intf = sym.integrate(f, x)\n",
    "intf"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 7,
   "id": "general-theme",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/latex": [
       "$\\displaystyle 27$"
      ],
      "text/plain": [
       "27"
      ]
     },
     "execution_count": 7,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "# The above examples are indefinite integrals of f.  We can evaluate\n",
    "# definite integrals using the same 'integrate()' function.  We just\n",
    "# need to pass the limits of integration...\n",
    "intf = sym.integrate(f, (x, 0, 3))\n",
    "intf"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 8,
   "id": "instrumental-rider",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/latex": [
       "$\\displaystyle 27$"
      ],
      "text/plain": [
       "27"
      ]
     },
     "execution_count": 8,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "# or:\n",
    "intf = f.integrate((x, 0, 3))\n",
    "intf"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 10,
   "id": "retired-variety",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/latex": [
       "$\\displaystyle - \\frac{2 a^{3}}{3} - 3 a + \\frac{2 b^{3}}{3} + 3 b$"
      ],
      "text/plain": [
       "-2*a**3/3 - 3*a + 2*b**3/3 + 3*b"
      ]
     },
     "execution_count": 10,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "# You can also used symbols for the integration limits.\n",
    "a = sym.Symbol('a')\n",
    "b = sym.Symbol('b')\n",
    "intf = f.integrate((x, a, b))\n",
    "intf"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 11,
   "id": "honest-douglas",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/latex": [
       "$\\displaystyle x \\sin{\\left(x y \\right)}$"
      ],
      "text/plain": [
       "x*sin(x*y)"
      ]
     },
     "execution_count": 11,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "# We can make our functions more complicated.  Here's a function of two variables.\n",
    "y = sym.Symbol('y')\n",
    "g = x*sym.sin(x*y)\n",
    "g"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 12,
   "id": "joint-ethernet",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/latex": [
       "$\\displaystyle \\begin{cases} - \\frac{x \\cos{\\left(x y \\right)}}{y} + \\frac{\\sin{\\left(x y \\right)}}{y^{2}} & \\text{for}\\: y \\neq 0 \\\\0 & \\text{otherwise} \\end{cases}$"
      ],
      "text/plain": [
       "Piecewise((-x*cos(x*y)/y + sin(x*y)/y**2, Ne(y, 0)), (0, True))"
      ]
     },
     "execution_count": 12,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "# Here's the x-integral of g...\n",
    "g.integrate(x)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 13,
   "id": "intense-elevation",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/latex": [
       "$\\displaystyle x \\left(\\begin{cases} - \\frac{\\cos{\\left(x y \\right)}}{x} & \\text{for}\\: x \\neq 0 \\\\0 & \\text{otherwise} \\end{cases}\\right)$"
      ],
      "text/plain": [
       "x*Piecewise((-cos(x*y)/x, Ne(x, 0)), (0, True))"
      ]
     },
     "execution_count": 13,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "# and here's the y-integral.\n",
    "g.integrate(y)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 14,
   "id": "thrown-large",
   "metadata": {},
   "outputs": [],
   "source": [
    "# Notice that the results of both of the g integrals depend on whether or not\n",
    "# x and y are zero.  We can specify that our variables are non-zero when they\n",
    "# are first declared.\n",
    "x = sym.Symbol('x', nonzero = True)\n",
    "y = sym.Symbol('y', nonzero = True)\n",
    "g = x*sym.sin(x*y)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 15,
   "id": "biological-nirvana",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "(-x*cos(x*y)/y + sin(x*y)/y**2, -cos(x*y))"
      ]
     },
     "execution_count": 15,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "# Now the solutions of our integrals have no conditions.\n",
    "g.integrate(x), g.integrate(y)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 16,
   "id": "compliant-production",
   "metadata": {},
   "outputs": [],
   "source": [
    "# Alternatively, you can define Python functions.\n",
    "def fcn(x):\n",
    "    return 2*x**2 + 3"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 17,
   "id": "dietary-department",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/latex": [
       "$\\displaystyle \\frac{2 x^{3}}{3} + 3 x$"
      ],
      "text/plain": [
       "2*x**3/3 + 3*x"
      ]
     },
     "execution_count": 17,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "# These functions can be integrated in the same way.  Here's the indefinite\n",
    "# integral...\n",
    "fcn(x).integrate(x)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 18,
   "id": "suited-bunny",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/latex": [
       "$\\displaystyle - \\frac{2 a^{3}}{3} - 3 a + \\frac{2 b^{3}}{3} + 3 b$"
      ],
      "text/plain": [
       "-2*a**3/3 - 3*a + 2*b**3/3 + 3*b"
      ]
     },
     "execution_count": 18,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "# ... and here's the indefinite integral.\n",
    "sym.integrate(fcn(x), (x, a, b))"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 19,
   "id": "welsh-disorder",
   "metadata": {},
   "outputs": [],
   "source": [
    "# Of course, you can have functions of multiple variables.\n",
    "def gcn(x, y):\n",
    "    return x*sym.sin(x*y)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 20,
   "id": "weighted-drink",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/latex": [
       "$\\displaystyle - \\frac{\\sin{\\left(x y \\right)}}{y}$"
      ],
      "text/plain": [
       "-sin(x*y)/y"
      ]
     },
     "execution_count": 20,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "# Here's the indefinite y-integral followed by the indefinite x-integral.\n",
    "z = sym.integrate(sym.integrate(gcn(x, y), y), x)\n",
    "z"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 21,
   "id": "higher-auditor",
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "-sin(1) + sin(3)/3 - sin(6)/3 + sin(2)\n",
      "0.208004944104050\n"
     ]
    }
   ],
   "source": [
    "# Here's the definite y-integral followed by the definite x-integral using\n",
    "# the other method of calling the 'integrate()' function.\n",
    "z = gcn(x, y).integrate((y, 1, 3)).integrate((x, -2, -1))\n",
    "print(z)\n",
    "print(sym.N(z))"
   ]
  }
 ],
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