symbolic_python.ipynb

{
 "cells": [
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "# Introduction\n",
    "\n",
    "The [SymPy website](https://www.sympy.org/en/index.html) states that SymPy is a Python library for symbolic mathematics. It aims to become a full-featured computer algebra system (CAS) while keeping the code as simple as possible in order to be comprehensible and easily extensible. SymPy is written entirely in Python.\n",
    "\n",
    "In this python notebook we will look at SymPy as it relates to:\n",
    "\n",
    "1. Setup, expressions, and methods\n",
    "2. Simple mathematical operators and solutions\n",
    "3. Limits\n",
    "4. Derivatives\n",
    "5. Basic antiderivatives (integrals)\n",
    "6. Initial value problems\n",
    "7. More complicated expressions\n",
    "8. Definite integrals\n",
    "9. Improper integrals\n",
    "10. Sequences and series\n",
    "\n",
    "Reference: [Mr. P Solver](https://www.youtube.com/watch?v=-SdIZHPuW9o&list=LL&index=1)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## 1. Import libraries and setup\n",
    "\n",
    "* Import all libraries from SymPy\n",
    "* Define symbol $x$ for use"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 36,
   "metadata": {},
   "outputs": [],
   "source": [
    "import sympy as smp\n",
    "from sympy import *\n",
    "import numpy as np\n",
    "import matplotlib.pyplot as plt\n",
    "\n",
    "x = smp.symbols('x')"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## 2. Simple mathematical operators and solutions\n",
    "Various mathematical operators can be used on defined symbols."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 37,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/latex": [
       "$\\displaystyle x^{2}$"
      ],
      "text/plain": [
       "x**2"
      ]
     },
     "execution_count": 37,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "x**2"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 38,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/latex": [
       "$\\displaystyle \\sin{\\left(x \\right)}$"
      ],
      "text/plain": [
       "sin(x)"
      ]
     },
     "execution_count": 38,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "smp.sin(x)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Expressions can be saved in new variables."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 39,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/latex": [
       "$\\displaystyle x^{2} + 4 x + 3$"
      ],
      "text/plain": [
       "x**2 + 4*x + 3"
      ]
     },
     "execution_count": 39,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "y = x**2 + 4*x + 3\n",
    "y"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Do operations on the new variables."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 40,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/latex": [
       "$\\displaystyle \\left(x^{2} + 4 x + 3\\right)^{2}$"
      ],
      "text/plain": [
       "(x**2 + 4*x + 3)**2"
      ]
     },
     "execution_count": 40,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "z = y**2\n",
    "z"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "We notice that the code above does not factorize. What needs to happen to the expression is decided by the different method called.\n",
    "Let's factorize the equation above."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 41,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/latex": [
       "$\\displaystyle \\left(x + 1\\right)^{2} \\left(x + 3\\right)^{2}$"
      ],
      "text/plain": [
       "(x + 1)**2*(x + 3)**2"
      ]
     },
     "execution_count": 41,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "smp.factor(z)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Next, let's solve the aforementioned equation using the `solve` method.\n",
    "\n",
    "`smp.solve(f, x)` simply finds the value(s) of $x$ that makes $f(x) = 0$."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 42,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "[-3, -1]"
      ]
     },
     "execution_count": 42,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "smp.solve(z, x)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 43,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "[0, pi]"
      ]
     },
     "execution_count": 43,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "smp.solve(smp.sin(x), x)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "**Good Practice:** If you know that the variable $x$ is real (and in some cases non-negatie) it is good to specify this when the variable is defined."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 44,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "[-I, I]"
      ]
     },
     "execution_count": 44,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "x = smp.symbols('x')\n",
    "smp.solve(x**2 + 1, x)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 13,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "[]"
      ]
     },
     "execution_count": 13,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "x = smp.symbols('x', real=True, positive=True)\n",
    "smp.solve(x**2 + 1, x)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "We can also define multible symbols/variables at the same time."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 45,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/latex": [
       "$\\displaystyle x^{2} + y \\sin{\\left(z \\right)}$"
      ],
      "text/plain": [
       "x**2 + y*sin(z)"
      ]
     },
     "execution_count": 45,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "x, y, z = smp.symbols('x y z')    # this works too smp.symbols('x,y,z')\n",
    "F = x**2 + smp.sin(z) * y\n",
    "F"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "We can also solve multivariate expressions, specifically which variable to solve. In each case, $F=0$ is being solved for."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 46,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "[-sqrt(-y*sin(z)), sqrt(-y*sin(z))]"
      ]
     },
     "execution_count": 46,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "x_sols = smp.solve(F, x)\n",
    "x_sols"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 47,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "[-x**2/sin(z)]"
      ]
     },
     "execution_count": 47,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "y_sols = smp.solve(F, y)\n",
    "y_sols"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 48,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "[asin(x**2/y) + pi, -asin(x**2/y)]"
      ]
     },
     "execution_count": 48,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "z_sols = smp.solve(F, z)\n",
    "z_sols"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Finally, converting a SumPy expression into a numerical function so that it can be plotted."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 49,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/latex": [
       "$\\displaystyle \\operatorname{asin}{\\left(\\frac{x^{2}}{y} \\right)} + \\pi$"
      ],
      "text/plain": [
       "asin(x**2/y) + pi"
      ]
     },
     "execution_count": 49,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "expr = z_sols[0]\n",
    "expr"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 50,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "3.665191429188092"
      ]
     },
     "execution_count": 50,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "expr_f = smp.lambdify([x,y], expr) # lambdify - Convert a SymPy expression into a function that allows for fast numeric evaluation.\n",
    "expr_f(1, 2) # lambdified function at a certian point"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 52,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": 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",
      "text/plain": [
       "<Figure size 432x288 with 1 Axes>"
      ]
     },
     "metadata": {
      "needs_background": "light"
     },
     "output_type": "display_data"
    }
   ],
   "source": [
    "x_num = np.linspace(0, 1, 100)\n",
    "y_num = 2\n",
    "plt.plot(x_num, expr_f(x_num, y_num))\n",
    "plt.show()"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "For multivariate expressions, values can be substituted in as well."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 53,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/latex": [
       "$\\displaystyle x^{2} + y \\sin{\\left(z \\right)}$"
      ],
      "text/plain": [
       "x**2 + y*sin(z)"
      ]
     },
     "execution_count": 53,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "F"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Substitute $y$ with $cos(z)$ and $z$ with $y$."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 55,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/latex": [
       "$\\displaystyle x^{2} + \\sin{\\left(y \\right)} \\cos{\\left(y \\right)}$"
      ],
      "text/plain": [
       "x**2 + sin(y)*cos(y)"
      ]
     },
     "execution_count": 55,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "F.subs([(y, smp.cos(z)), (z, y)]) # but it's not assigned?"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "3. Limits"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "$$\\lim_{x \\to \\pi}\\sin\\left(\\frac{x}{2}+\\sin(x)\\right)$$"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 56,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/latex": [
       "$\\displaystyle 1$"
      ],
      "text/plain": [
       "1"
      ]
     },
     "execution_count": 56,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "smp.limit(smp.sin(x/2 + smp.sin(x)), x, smp.pi)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "$$\\lim_{x \\to 0^+}\\frac{2e^{1/x}}{e^{1/x}+1}$$"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 58,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/latex": [
       "$\\displaystyle 2$"
      ],
      "text/plain": [
       "2"
      ]
     },
     "execution_count": 58,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "smp.limit((2*smp.exp(1/x))/(smp.exp(1/x)+1), x, 0, dir='+')"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "$$\\lim_{x \\to 0^-}\\frac{2e^{1/x}}{e^{1/x}+1}$$"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 59,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/latex": [
       "$\\displaystyle 0$"
      ],
      "text/plain": [
       "0"
      ]
     },
     "execution_count": 59,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "smp.limit((2*smp.exp(1/x))/(smp.exp(1/x)+1), z=x, z0=0, dir='-')"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "$$\\lim_{x \\to \\infty}\\frac{\\cos{x}-1}{x}$$"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 61,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/latex": [
       "$\\displaystyle 0$"
      ],
      "text/plain": [
       "0"
      ]
     },
     "execution_count": 61,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "smp.limit((smp.cos(x)-1)/(x), z=x, z0=smp.oo)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## 4. Derivatives\n",
    "$$\\frac{d}{dx}\\left(\\frac{1+\\sin{x}}{1-\\cos{x}}\\right)^2$$"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 65,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/latex": [
       "$\\displaystyle \\frac{2 \\left(\\sin{\\left(x \\right)} + 1\\right) \\cos{\\left(x \\right)}}{\\left(1 - \\cos{\\left(x \\right)}\\right)^{2}} - \\frac{2 \\left(\\sin{\\left(x \\right)} + 1\\right)^{2} \\sin{\\left(x \\right)}}{\\left(1 - \\cos{\\left(x \\right)}\\right)^{3}}$"
      ],
      "text/plain": [
       "2*(sin(x) + 1)*cos(x)/(1 - cos(x))**2 - 2*(sin(x) + 1)**2*sin(x)/(1 - cos(x))**3"
      ]
     },
     "execution_count": 65,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "smp.diff(((1+smp.sin(x))/(1-smp.cos(x)))**2, x)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "$$\\frac{d}{dx}\\left(\\log_{5}{x}\\right)^{\\frac{x}{2}}$$"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 66,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/latex": [
       "$\\displaystyle \\left(\\frac{\\log{\\left(x \\right)}}{\\log{\\left(5 \\right)}}\\right)^{\\frac{x}{2}} \\left(\\frac{\\log{\\left(\\frac{\\log{\\left(x \\right)}}{\\log{\\left(5 \\right)}} \\right)}}{2} + \\frac{1}{2 \\log{\\left(x \\right)}}\\right)$"
      ],
      "text/plain": [
       "(log(x)/log(5))**(x/2)*(log(log(x)/log(5))/2 + 1/(2*log(x)))"
      ]
     },
     "execution_count": 66,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "smp.diff((smp.log(x,5))**(x/2))"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "$$\\frac{d}{dx}f\\left(x+g(x)\\right)$$"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 71,
   "metadata": {},
   "outputs": [],
   "source": [
    "f, g = smp.symbols(\"f g\", cls=smp.Function)\n",
    "g = g(x)\n",
    "f = f(x+g)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 72,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/latex": [
       "$\\displaystyle \\left(\\frac{d}{d x} g{\\left(x \\right)} + 1\\right) \\left. \\frac{d}{d \\xi_{1}} f{\\left(\\xi_{1} \\right)} \\right|_{\\substack{ \\xi_{1}=x + g{\\left(x \\right)} }}$"
      ],
      "text/plain": [
       "(Derivative(g(x), x) + 1)*Subs(Derivative(f(_xi_1), _xi_1), _xi_1, x + g(x))"
      ]
     },
     "execution_count": 72,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "smp.diff(f, x)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## 5. Basic antiderivitavies (integrals)\n",
    "$$\\int\\csc(x)\\cot(x)dx$$"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 73,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/latex": [
       "$\\displaystyle - \\frac{1}{\\sin{\\left(x \\right)}}$"
      ],
      "text/plain": [
       "-1/sin(x)"
      ]
     },
     "execution_count": 73,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "smp.integrate((smp.csc(x)*smp.cot(x)), x)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "$$\\int\\left(\\frac{2}{\\sqrt{1-x^2}}-\\frac{1}{x^{1/4}}\\right)dx$$"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 76,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/latex": [
       "$\\displaystyle - \\frac{4 x^{\\frac{3}{4}}}{3} + 2 \\operatorname{asin}{\\left(x \\right)}$"
      ],
      "text/plain": [
       "-4*x**(3/4)/3 + 2*asin(x)"
      ]
     },
     "execution_count": 76,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "smp.integrate((2/smp.sqrt(1-x**2))-(1/x**smp.Rational(1,4)),x)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## 6. Initial value problems\n",
    "Given $\\frac{dy}{dx}=8x+\\csc^2(x)$ with $y(\\pi/2)=-7$\n",
    "\n",
    "Solve for $y(x)$"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 78,
   "metadata": {},
   "outputs": [],
   "source": [
    "integral = smp.integrate(8*x+smp.csc(x)**2, x)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 79,
   "metadata": {},
   "outputs": [],
   "source": [
    "c = - integral.subs(x, smp.pi/2) - 7\n",
    "y = integral + c"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 80,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/latex": [
       "$\\displaystyle -7$"
      ],
      "text/plain": [
       "-7"
      ]
     },
     "execution_count": 80,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "y.subs(x, smp.pi/2)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 81,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/latex": [
       "$\\displaystyle 4 x^{2} - \\pi^{2} - 7 - \\frac{\\cos{\\left(x \\right)}}{\\sin{\\left(x \\right)}}$"
      ],
      "text/plain": [
       "4*x**2 - pi**2 - 7 - cos(x)/sin(x)"
      ]
     },
     "execution_count": 81,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "y"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## 7. More complicated expressions\n",
    "$$\\int\\frac{(1+\\sqrt{x})^{1/3}}{\\sqrt{x}}dx$$"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 85,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/latex": [
       "$\\displaystyle \\frac{3 \\sqrt{x} \\sqrt[3]{\\sqrt{x} + 1}}{2} + \\frac{3 \\sqrt[3]{\\sqrt{x} + 1}}{2}$"
      ],
      "text/plain": [
       "3*sqrt(x)*(sqrt(x) + 1)**(1/3)/2 + 3*(sqrt(x) + 1)**(1/3)/2"
      ]
     },
     "execution_count": 85,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "smp.integrate(((1+smp.sqrt(x))**smp.Rational(1,3))/(smp.sqrt(x)), x)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "$$$$"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "$$\\int x(1-x^2)^{1/4}dx$$"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 83,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/latex": [
       "$\\displaystyle \\frac{2 x^{2} \\sqrt[4]{1 - x^{2}}}{5} - \\frac{2 \\sqrt[4]{1 - x^{2}}}{5}$"
      ],
      "text/plain": [
       "2*x**2*(1 - x**2)**(1/4)/5 - 2*(1 - x**2)**(1/4)/5"
      ]
     },
     "execution_count": 83,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "smp.integrate(x*(1-x**2)**smp.Rational(1,4), x)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "$$\\int\\frac{(2x-1)\\cos{(\\sqrt{3(2x-1)^2+6})}}{\\sqrt{3(2x-1)^2+6}}dx$$"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 84,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/latex": [
       "$\\displaystyle \\frac{\\sin{\\left(\\sqrt{3 \\left(2 x - 1\\right)^{2} + 6} \\right)}}{6}$"
      ],
      "text/plain": [
       "sin(sqrt(3*(2*x - 1)**2 + 6))/6"
      ]
     },
     "execution_count": 84,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "smp.integrate((2*x-1)*(smp.cos(smp.sqrt(3*(2*x-1)**2+6)))/(smp.sqrt(3*(2*x-1)**2+6)), x)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## 9. Definite integrals\n",
    "$$\\int_0^{ln(4)}\\frac{e^x}{\\sqrt{e^{2x}+9}}dx$$"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 86,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/latex": [
       "$\\displaystyle - \\operatorname{asinh}{\\left(\\frac{1}{3} \\right)} + \\operatorname{asinh}{\\left(\\frac{4}{3} \\right)}$"
      ],
      "text/plain": [
       "-asinh(1/3) + asinh(4/3)"
      ]
     },
     "execution_count": 86,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "smp.integrate((smp.exp(x))/(smp.sqrt(smp.exp(2*x)+9)), (x, 0, smp.log(4)))"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "$$\\int_1^tx^{10}e^{x}dx$$"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 88,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/latex": [
       "$\\displaystyle \\left(t^{10} - 10 t^{9} + 90 t^{8} - 720 t^{7} + 5040 t^{6} - 30240 t^{5} + 151200 t^{4} - 604800 t^{3} + 1814400 t^{2} - 3628800 t + 3628800\\right) e^{t} - 1334961 e$"
      ],
      "text/plain": [
       "(t**10 - 10*t**9 + 90*t**8 - 720*t**7 + 5040*t**6 - 30240*t**5 + 151200*t**4 - 604800*t**3 + 1814400*t**2 - 3628800*t + 3628800)*exp(t) - 1334961*E"
      ]
     },
     "execution_count": 88,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "t = smp.symbols('t')\n",
    "smp.integrate(x**10*smp.exp(x), (x, 1, t))"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## 10. Improper integrals\n",
    "$$\\int_0^\\infty\\frac{16\\tan^{-1}{(x)}}{1+x^2}dx$$"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 89,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/latex": [
       "$\\displaystyle 2 \\pi^{2}$"
      ],
      "text/plain": [
       "2*pi**2"
      ]
     },
     "execution_count": 89,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "smp.integrate((16*smp.atan(x))/(1+x**2), (x, 0, smp.oo))"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## 11. Sequences and series\n",
    "$$\\sum_{n=0}^{\\infty}\\frac{6}{4^n}$$"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 91,
   "metadata": {},
   "outputs": [],
   "source": [
    "n = smp.symbols('n')"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 92,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/latex": [
       "$\\displaystyle 8$"
      ],
      "text/plain": [
       "8"
      ]
     },
     "execution_count": 92,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "smp.summation(6/4**n, (n, 0, smp.oo))"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "$$\\sum_{n=0}^{\\infty}\\frac{2^{n+1}}{5^n}$$"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 93,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/latex": [
       "$\\displaystyle \\frac{10}{3}$"
      ],
      "text/plain": [
       "10/3"
      ]
     },
     "execution_count": 93,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "smp.summation((2**(n+1))/(5**n), (n, 0, smp.oo))"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "$$\\sum_{n=0}^{\\infty}\\frac{tan^{-1}(n)}{n^{1.1}}$$"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 96,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/latex": [
       "$\\displaystyle \\sum_{n=0}^{\\infty} \\frac{\\operatorname{atan}{\\left(n \\right)}}{n^{1.1}}$"
      ],
      "text/plain": [
       "Sum(atan(n)/n**1.1, (n, 0, oo))"
      ]
     },
     "execution_count": 96,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "smp.summation(smp.atan(n)/n**(1.1), (n, 0, smp.oo)).doit()"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "$$\\sum_{n=0}^{\\infty}\\frac{1+\\cos{x}}{n^2}$$"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 100,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/latex": [
       "$\\displaystyle \\sum_{n=0}^{\\infty} \\frac{\\cos{\\left(n \\right)} + 1}{n^{2}}$"
      ],
      "text/plain": [
       "Sum((cos(n) + 1)/n**2, (n, 0, oo))"
      ]
     },
     "execution_count": 100,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "smp.summation((1+smp.cos(n))/n**2, (n, 0, smp.oo)).doit()"
   ]
  }
 ],
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