{
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  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": [
    "from sympy import *\n",
    "init_printing(use_latex='mathjax')\n",
    "n, m = symbols('n,m', integer=True)\n",
    "x, y, z = symbols('x,y,z')"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Integrals\n",
    "\n",
    "In the last section we learned symbolic differentiation with `.diff`.  Here we'll cover symbolic integration with `integrate`."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Here is how we write the indefinite integral\n",
    "\n",
    "$$ \\int x^2 dx = \\frac{x^3}{3}$$"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": [
    "# Indefinite integral\n",
    "integrate(x**2, x)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "And the definite integral\n",
    "\n",
    "$$ \\int_0^3 x^2 dx = \\left.\\frac{x^3}{3} \\right|_0^3 = \\frac{3^3}{3} - \\frac{0^3}{3} = 9 $$"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": [
    "# Definite integral\n",
    "integrate(x**2, (x, 0, 3))"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "As always, because we're using symbolics, we could use a symbol whenever we previously used a number\n",
    "\n",
    "$$ \\int_y^z x^n dx $$"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": [
    "integrate(x**n, (x, y, z))"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### Exercise\n",
    "\n",
    "Compute the following integrals:\n",
    "\n",
    "$$ \\int \\sin(x) dx $$\n",
    "$$ \\int_0^{\\pi} \\sin(x) dx $$\n",
    "$$ \\int_0^y x^5 + 12x^3 - 2x + 1  $$\n",
    "$$ \\int e^{\\frac{(x - \\mu)^2}{\\sigma^2}} $$\n",
    "\n",
    "Feel free to play with various parameters and settings and see how the results change."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": [
    "# Use `integrate` to solve the integrals above\n",
    "\n"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Are there some integrals that SymPy can't do?  Find some."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": [
    "# Use `integrate` on other equations.  Symbolic integration has it limits, find them."
   ]
  }
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