{
 "cells": [
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "# Chapter 6 : Integration And Its Applications"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Example 6.1.1, page no. 199"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 37,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "Indefinite integral\n",
      "3*x/8 - sin(x)**3*cos(x)/4 - 3*sin(x)*cos(x)/8\n"
     ]
    }
   ],
   "source": [
    "import sympy,numpy\n",
    "\n",
    "print \"Indefinite integral\"\n",
    "x = sympy.Symbol('x')\n",
    "f = sympy.integrate((sympy.sin(x))**4,(x))\n",
    "print f"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Example 6.1.2, page no. 200"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 39,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "Indefinite integral\n",
      "-sin(x)**7/7 + 3*sin(x)**5/5 - sin(x)**3 + sin(x)\n"
     ]
    }
   ],
   "source": [
    "import sympy,numpy\n",
    "\n",
    "print \"Indefinite integral\"\n",
    "x = sympy.Symbol('x')\n",
    "f = sympy.integrate((sympy.cos(x))**7,(x))\n",
    "print f"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Example 6.2.1, page no. 202"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 41,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "Definite integral\n",
      "0.490873852123\n"
     ]
    }
   ],
   "source": [
    "import sympy,numpy,math\n",
    "\n",
    "print \"Definite integral\"\n",
    "x = sympy.Symbol('x')\n",
    "f = sympy.integrate((sympy.cos(x))**6,(x,0,math.pi/2))\n",
    "print float(f)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Example 6.2.2, page no. 202"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 43,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "Definite integral\n",
      "zoo*a**6 + a**6*log(-2*a**2)/4 - 11*a**6/24\n"
     ]
    }
   ],
   "source": [
    "import sympy,numpy,math\n",
    "\n",
    "print \"Definite integral\"\n",
    "x = sympy.Symbol('x')\n",
    "a = sympy.Symbol('a')\n",
    "g = x**7/(a**2-x**2)**1/2\n",
    "f = sympy.integrate(g,(x,0,a))\n",
    "print f"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Example 6.2.3, page no. 203"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 45,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "Definite integral\n",
      "4*a**4\n"
     ]
    }
   ],
   "source": [
    "import sympy,numpy,math\n",
    "\n",
    "print \"Definite integral\"\n",
    "x = sympy.Symbol('x')\n",
    "a = sympy.Symbol('a')\n",
    "g = x**3*(2*a*x-x**2)**(1/2) \n",
    "f = sympy.integrate(g,(x,0,2*a))\n",
    "print f"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Example 6.2.4, page no. 204"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 23,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "Definite integral\n",
      "0.5*(a**2 + 0.5625)**(-n) + 0.5*(a**2 + 0.0625)**(-n)\n"
     ]
    }
   ],
   "source": [
    "from sympy.abc import x,a,n\n",
    "from sympy.integrals import Integral\n",
    "print \"Definite integral\"\n",
    "g = 1./(a**2+x**2)**n \n",
    "f = Integral(g,(x,0,1))\n",
    "print f.as_sum(2).n()"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Example 6.4.1, page no. 207"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 4,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "Definite integral\n",
      "0.0532891345937363\n"
     ]
    }
   ],
   "source": [
    "from sympy.abc import x,a,n\n",
    "from sympy.integrals import Integral\n",
    "import sympy, math\n",
    "print \"Definite integral\"\n",
    "g = (sympy.sin(6*x))**3*(sympy.cos(3*x))**7\n",
    "f = Integral(g,(x,0,math.pi/6))\n",
    "print f.as_sum(2).n()"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Example 6.4.2, page no. 208"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 52,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "Definite integral\n",
      "0.0571428571429\n"
     ]
    }
   ],
   "source": [
    "import sympy,numpy,math\n",
    "\n",
    "print \"Definite integral\"\n",
    "x = sympy.Symbol('x')\n",
    "g = (sympy.sin(6*x))**3*(sympy.cos(3*x))**7\n",
    "g = x**4*(1-x**2)**(3/2)\n",
    "f = sympy.integrate(g,(x,0,1)) \n",
    "print float(f)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Example 6.5, page no. 208"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 63,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "Definite integral\n",
      "Enter n : 0\n",
      "Enter m : 1\n",
      "1.0\n",
      "1.0\n",
      "Equal\n"
     ]
    }
   ],
   "source": [
    "import sympy,numpy,math\n",
    "\n",
    "print \"Definite integral\"\n",
    "x = sympy.Symbol('x')\n",
    "m = sympy.Symbol('m')\n",
    "n = sympy.Symbol('n')\n",
    "n = int(raw_input(\"Enter n : \"))\n",
    "m = int(raw_input(\"Enter m : \"))\n",
    "g =(sympy.cos(x))**m*sympy.cos(n*x)\n",
    "f = sympy.integrate(g,(x,0,math.pi/2))\n",
    "print float(f) \n",
    "g2 =(sympy.cos(x))**(m-1)*sympy.cos((n-1)*x)\n",
    "f2 = m/(m+n)*sympy.integrate(g2,(x,0,math.pi/2))\n",
    "print float(f2)\n",
    "print \"Equal\""
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Example 6.6.1, page no. 210"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 65,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "Definite integral\n",
      "Enter n : 1\n",
      "Piecewise((x*exp(-I*x)*sin(x)/2 - I*x*exp(-I*x)*cos(x)/2 - exp(-I*x)*cos(x)/2, a == -I), (x*exp(I*x)*sin(x)/2 + I*x*exp(I*x)*cos(x)/2 - exp(I*x)*cos(x)/2, a == I), (a*exp(a*x)*sin(x)/(a**2 + 1) - exp(a*x)*cos(x)/(a**2 + 1), True))\n"
     ]
    }
   ],
   "source": [
    "import sympy,numpy,math\n",
    "\n",
    "print \"Definite integral\"\n",
    "x = sympy.Symbol('x')\n",
    "a = sympy.Symbol('a')\n",
    "n = int(raw_input(\"Enter n : \"))\n",
    "g = sympy.exp(a*x)*(sympy.sin(x))**n\n",
    "f = sympy.integrate(g,(x))\n",
    "print f"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Example 6.7.1, page no. 212"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 66,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "(4*sin(x)**2 - 3)/(4*sin(x)**4 - 8*sin(x)**2 + 4) - log(sin(x)**2 - 1)/2\n"
     ]
    }
   ],
   "source": [
    "import sympy,numpy,math\n",
    "\n",
    "x = sympy.Symbol('x')\n",
    "print sympy.integrate(sympy.tan(x)**5,(x))"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Example 6.8, page no. 215"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 11,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "Enter the value of n : 3\n",
      "n(p+q)=  0.999999999999999\n"
     ]
    }
   ],
   "source": [
    "import numpy,sympy,math\n",
    "\n",
    "n = int(raw_input(\"Enter the value of n : \"))\n",
    "p = sympy.integrate((sympy.tan(x))**(n-1),('x',0,math.pi/4))\n",
    "q = sympy.integrate((sympy.tan(x))**(n+1),('x',0,math.pi/4))\n",
    "print \"n(p+q)= \",\n",
    "print n*(p+q)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Example 6.9.1, page no. 217"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 69,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "1.33333333333333\n"
     ]
    }
   ],
   "source": [
    "import sympy,numpy,math\n",
    "\n",
    "x = sympy.Symbol('x')\n",
    "g = sympy.sec(x)**4\n",
    "print sympy.integrate(g,('x',0,math.pi/4))"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Example 6.9.2, page no. 217"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 36,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "0.607986405500361\n"
     ]
    }
   ],
   "source": [
    "import sympy,numpy,math\n",
    "\n",
    "x = sympy.Symbol('x')\n",
    "print sympy.integrate('1/sin(x)**3',('x',math.pi/3,math.pi/2))"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Example 6.10, page no. 220"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 20,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "0.000362418686373\n"
     ]
    }
   ],
   "source": [
    "import sympy,math\n",
    "from sympy.abc import x\n",
    "from sympy.integrals import Integral\n",
    "import sympy, math\n",
    "g = x*sympy.sin(x)**6*sympy.cos(x)**4\n",
    "f = Integral(g,(x,0,math.pi/6))\n",
    "print float(f)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Example 6.12, page no. 221"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 17,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "0.785398163397\n"
     ]
    }
   ],
   "source": [
    "import sympy,math\n",
    "from sympy.abc import x\n",
    "from sympy.integrals import Integral\n",
    "x = sympy.Symbol('x')\n",
    "a = math.pi/2\n",
    "f = (sympy.sin(x)**0.5)/(sympy.sin(x)**0.5+sympy.cos(x)**0.5)\n",
    "f = Integral(f,(x, 0, a))\n",
    "print float(f)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Example 6.13, page no. 223"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 4,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "The summation is equivalent to integration of 1/(1+xˆ2) from 0 to 1\n",
      "0.785398163397\n"
     ]
    }
   ],
   "source": [
    "import sympy,numpy,math\n",
    "\n",
    "x = sympy.Symbol('x')\n",
    "print \"The summation is equivalent to integration of 1/(1+xˆ2) from 0 to 1\"\n",
    "g = 1/(1+x**2)\n",
    "f = sympy.integrate(g,(x,0,1))\n",
    "print float(f)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Example 6.14, page no.  223"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 5,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "The summation is equivalent to integration of log(1+x) from 0 to 1\n",
      "0.38629436112\n"
     ]
    }
   ],
   "source": [
    "import sympy,numpy,math\n",
    "\n",
    "x = sympy.Symbol('x')\n",
    "print \"The summation is equivalent to integration of log(1+x) from 0 to 1\"\n",
    "g = sympy.log(1+x)\n",
    "f = sympy.integrate(g,(x,0,1))\n",
    "print float(f) "
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Example 6.15, page no. 225"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 31,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "0.0337339994178\n"
     ]
    }
   ],
   "source": [
    "import sympy,math\n",
    "from sympy.integrals import Integral\n",
    "f = Integral(x*sympy.sin(x)**8*sympy.cos(x)**4,(x,0,math.pi))\n",
    "print float(f)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Example 6.16, page no. 226"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 30,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "-1.08879304515\n"
     ]
    }
   ],
   "source": [
    "import numpy,sympy,math\n",
    "from sympy.integrals import Integral\n",
    "x = sympy.Symbol('x')\n",
    "#f = sympy.integrate(sympy.log(sympy.sin(x)),('x',0,math.pi/2))\n",
    "f = Integral(sympy.log(sympy.sin(x)), (x, 0, 1.5707963267949))\n",
    "print float(f)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Example 6.24, page no. 234"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 20,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "from the graph, it is clear that the points of intersection are x=−4 and x =8.\n",
      "So, our region of integration is from x=−4 to x=8\n",
      "36\n"
     ]
    },
    {
     "data": {
      "image/png": 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Y3TfzYs6uOUzZMoUxXcdQq3wtIHwHevPFha4ABgJ/U9Ut+blD993EN6vqPwBU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\nlWV1AAAAAElFTkSuQmCC\n",
      "text/plain": [
       "<matplotlib.figure.Figure at 0x7faae1cad790>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "%matplotlib inline\n",
    "import matplotlib.pyplot as plt\n",
    "import numpy\n",
    "import math\n",
    "\n",
    "x = numpy.linspace(-5,10,70)\n",
    "y1 = (x+8)/2\n",
    "y2 = x**2/8\n",
    "\n",
    "plt.xlabel('X axis')\n",
    "plt.ylabel('Y axis')\n",
    "plt.title('My Graph')\n",
    "plt.plot(x, y1, \"o-\" )\n",
    "plt.plot(x, y2, \"+-\" )\n",
    "plt.legend([\"(x+8)/2\",\"x**2/8\"])\n",
    "print \"from the graph, it is clear that the points of intersection are x=−4 and x =8.\"\n",
    "print \"So, our region of integration is from x=−4 to x=8\"\n",
    "print sympy.integrate('(x+8)/2-x**2/8',('x',-4,8))"
   ]
  }
 ],
 "metadata": {
  "kernelspec": {
   "display_name": "Python 2",
   "language": "python",
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