2017年7月11日火曜日

学習環境

解析入門〈2〉(松坂 和夫(著)、岩波書店)の第8章(積分の計算)、8.1(不定積分の計算)、問題5.を取り組んでみる。


    1. sin 3 x cos 3 x dx =I( 3,3 ) = sin 4 x cos 2 x 2 + 33+2 2 I( 3,1 ) = sin 4 x cos 2 x 2 ( sin 2 x cos 0 x 2 + 2 2 I( 1,1 ) ) = 1 2 tan 2 x sin 2 x+ sin 2 x 2 sinx cos 1 xdx = 1 2 tan 2 x sin 2 x+ sin 2 x 2 +log| cosx |

    2. I( 2,2 ) = sin 3 xcosx 4 + 1 4 I( 2,0 ) = sin 3 xcosx 4 + 1 4 sin 2 xdx sin 2 xdx = sinx d dx ( cosx )dx =sinxcosx+ cos 2 xdx =sinxcosx+ ( 1 sin 2 x )dx =sinxcosx+x sin 2 xdx sin 2 xdx = 1 2 ( xsinxcosx ) I( 2,2 ) = sin 3 xcosx 4 + 1 4 1 2 ( xsinxcosx ) = sin 3 xcosx 4 + 1 8 ( xsinxcosx )

    3. I( 3,3 ) = sin 2 x cos 2 x 2 + 4 2 I( 3,1 ) = sin 2 x cos 2 x 2 +2( sin 2 x cos 0 x 2 + 2 2 I( 1,1 ) ) = sin 2 x cos 2 x 2 sin 2 x+2I( 1,1 ) = sin 2 x cos 2 x 2 sin 2 x+2 1 sinxcosx dx 1 sinxcosx dx cosx sinx · 1 cos 2 x dx =log| tanx | I( 3,3 )= sin 2 x cos 2 x 2 sin 2 x+2log| tanx |

コード(Emacs)

Python 3

#!/usr/bin/env python3
# -*- coding: utf-8 -*-

from sympy import pprint, symbols, Integral, sin, cos, tan

print('5.')
x = symbols('x')

fs = [tan(x) ** 3,
      sin(x) ** 2 * cos(x) ** 2,
      1 / (sin(x) ** 3 * cos(x) ** 3)]

for i, f in enumerate(fs, 1):
    I = Integral(f, x)
    pprint(I)
    I = I.doit()
    pprint(I)
    pprint(I.factor())
    pprint(I.expand())

入出力結果(Terminal, IPython)

$ ./sample5.py
5.
⌠           
⎮    3      
⎮ tan (x) dx
⌡           
   ⎛   2       ⎞                
log⎝sin (x) - 1⎠         1      
──────────────── - ─────────────
       2                2       
                   2⋅sin (x) - 2
   ⎛   2       ⎞    2         ⎛   2       ⎞    
log⎝sin (x) - 1⎠⋅sin (x) - log⎝sin (x) - 1⎠ - 1
───────────────────────────────────────────────
          2⋅(sin(x) - 1)⋅(sin(x) + 1)          
   ⎛   2       ⎞                
log⎝sin (x) - 1⎠         1      
──────────────── - ─────────────
       2                2       
                   2⋅sin (x) - 2
⌠                   
⎮    2       2      
⎮ sin (x)⋅cos (x) dx
⌡                   
x   sin(2⋅x)⋅cos(2⋅x)
─ - ─────────────────
8           16       
2⋅x - sin(2⋅x)⋅cos(2⋅x)
───────────────────────
           16          
x   sin(2⋅x)⋅cos(2⋅x)
─ - ─────────────────
8           16       
⌠                   
⎮        1          
⎮ ─────────────── dx
⎮    3       3      
⎮ sin (x)⋅cos (x)   
⌡                   
           2                                              
      2⋅sin (x) - 1          ⎛   2       ⎞                
- ───────────────────── - log⎝sin (x) - 1⎠ + 2⋅log(sin(x))
       4           2                                      
  2⋅sin (x) - 2⋅sin (x)                                   
       ⎛   2       ⎞    4           ⎛   2       ⎞    2                       4
- 2⋅log⎝sin (x) - 1⎠⋅sin (x) + 2⋅log⎝sin (x) - 1⎠⋅sin (x) + 4⋅log(sin(x))⋅sin 
──────────────────────────────────────────────────────────────────────────────
                                                                          2   
                                           2⋅(sin(x) - 1)⋅(sin(x) + 1)⋅sin (x)

                       2           2       
(x) - 4⋅log(sin(x))⋅sin (x) - 2⋅sin (x) + 1
───────────────────────────────────────────
                                           
                                           
                                                2                             
     ⎛   2       ⎞                         2⋅sin (x)                   1      
- log⎝sin (x) - 1⎠ + 2⋅log(sin(x)) - ───────────────────── + ─────────────────
                                          4           2           4           
                                     2⋅sin (x) - 2⋅sin (x)   2⋅sin (x) - 2⋅sin

    
    
────
2   
 (x)
$

HTML5

<div id="graph0"></div>
<pre id="output0"></pre>
<label for="r0">r = </label>
<input id="r0" type="number" min="0" value="0.5">
<label for="dx">dx = </label>
<input id="dx" type="number" min="0" step="0.0001" value="0.001">
<br>
<label for="x1">x1 = </label>
<input id="x1" type="number" value="-5">
<label for="x2">x2 = </label>
<input id="x2" type="number" value="5">
<br>
<label for="y1">y1 = </label>
<input id="y1" type="number" value="-5">
<label for="y2">y2 = </label>
<input id="y2" type="number" value="5">

<button id="draw0">draw</button>
<button id="clear0">clear</button>

<script type="text/javascript" src="https://cdnjs.cloudflare.com/ajax/libs/d3/4.2.6/d3.min.js" integrity="sha256-5idA201uSwHAROtCops7codXJ0vja+6wbBrZdQ6ETQc=" crossorigin="anonymous"></script>

<script src="sample5.js"></script>    

JavaScript

let div0 = document.querySelector('#graph0'),
    pre0 = document.querySelector('#output0'),
    width = 600,
    height = 600,
    padding = 50,
    btn0 = document.querySelector('#draw0'),
    btn1 = document.querySelector('#clear0'),
    input_r = document.querySelector('#r0'),
    input_dx = document.querySelector('#dx'),
    input_x1 = document.querySelector('#x1'),
    input_x2 = document.querySelector('#x2'),
    input_y1 = document.querySelector('#y1'),
    input_y2 = document.querySelector('#y2'),
    inputs = [input_r, input_dx, input_x1, input_x2, input_y1, input_y2],
    p = (x) => pre0.textContent += x + '\n',
    range = (start, end, step=1) => {
        let res = [];
        for (let i = start; i < end; i += step) {
            res.push(i);
        }
        return res;
    };

let f1 = (x) => Math.tan(x) ** 3,
    f2 = (x) => Math.sin(x) ** 2 * Math.cos(x) ** 2,
    f3 = (x) => 1 / (Math.sin(x) ** 3 * Math.cos(x) ** 3);

let draw = () => {
    pre0.textContent = '';

    let r = parseFloat(input_r.value),
        dx = parseFloat(input_dx.value),
        x1 = parseFloat(input_x1.value),
        x2 = parseFloat(input_x2.value),
        y1 = parseFloat(input_y1.value),
        y2 = parseFloat(input_y2.value);
    
    if (r === 0 || dx === 0 || x1 > x2 || y1 > y2) {
        return;
    }

    let points = [],
        lines = [],
        fns = [[f1, 'red'],
               [f2, 'green'],
               [f3, 'blue']];

    fns
        .forEach((o) => {
            let [fn, color] = o;
            
            for (let x = x1; x <= x2; x += dx) {
                let y = fn(x);
                
                if (Math.abs(y) < Infinity) {
                    points.push([x, y, color]);
                }
            }
        });
    
    let xscale = d3.scaleLinear()
        .domain([x1, x2])
        .range([padding, width - padding]);

    let yscale = d3.scaleLinear()
        .domain([y1, y2])
        .range([height - padding, padding]);

    let xaxis = d3.axisBottom().scale(xscale);
    let yaxis = d3.axisLeft().scale(yscale);
    div0.innerHTML = '';
    let svg = d3.select('#graph0')
        .append('svg')
        .attr('width', width)
        .attr('height', height);

    svg.selectAll('circle')
        .data(points)
        .enter()
        .append('circle')
        .attr('cx', (d) => xscale(d[0]))
        .attr('cy', (d) => yscale(d[1]))
        .attr('r', r)
        .attr('fill', (d) => d[2] || 'green');

    svg.selectAll('line')
        .data([[x1, 0, x2, 0], [0, y1, 0, y2]].concat(lines))
        .enter()
        .append('line')
        .attr('x1', (d) => xscale(d[0]))
        .attr('y1', (d) => yscale(d[1]))
        .attr('x2', (d) => xscale(d[2]))
        .attr('y2', (d) => yscale(d[3]))
        .attr('stroke', (d) => d[4] || 'black');
    
    svg.append('g')
        .attr('transform', `translate(0, ${height - padding})`)
        .call(xaxis);

    svg.append('g')
        .attr('transform', `translate(${padding}, 0)`)
        .call(yaxis);
    p(fns.join('\n'));
};

inputs.forEach((input) => input.onchange = draw);
btn0.onclick = draw;
btn1.onclick = () => pre0.textContent = '';
draw();







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