{
 "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",
    "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": "iVBORw0KGgoAAAANSUhEUgAAAYAAAAEZCAYAAACervI0AAAABHNCSVQICAgIfAhkiAAAAAlwSFlz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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",
   "name": "python2"
  },
  "language_info": {
   "codemirror_mode": {
    "name": "ipython",
    "version": 2
   },
   "file_extension": ".py",
   "mimetype": "text/x-python",
   "name": "python",
   "nbconvert_exporter": "python",
   "pygments_lexer": "ipython2",
   "version": "2.7.10"
  }
 },
 "nbformat": 4,
 "nbformat_minor": 0
}