2017年8月23日水曜日

学習環境

数学読本〈5〉微分法の応用/積分法/積分法の応用/行列と行列式(松坂 和夫(著)、岩波書店)の第19章(細分による加法 - 積分法)、19.2(不定積分の計算)、分数関数の積分、問19.を取り組んでみる。


    1. log( 1+x ) x 2 dx = 1 x log( 1+x ) 1 x( 1+x ) dx = log( 1+x ) x + 1 x( x+1 ) dx a x + b x+1 = ( a+b )x+a x( x+1 ) a+b=0 a=1 b=1 log( 1+x ) x + ( 1 x 1 x+1 )dx = log( 1+x ) x +log| x |log( x+1 )

    2. xlog( x 2 +1 )dx = 1 2 x 2 log( x 2 +1 ) 2 x 2 x 2 +1 dx = x 2 log( x 2 +1 ) 2 2( x 2 +1 )2 x 2 +1 dx = x 2 log( x 2 +1 ) 2 ( 2dx2 1 x 2 +1 dx ) = x 2 log( x 2 +1 ) 2 2x+2arctanx

    3. e x =t e x = dt dx 1 t+1 1 t dt = 1 t( t+1 ) dt a t + b t+1 = ( a+b )t+a t( t+1 ) a+b=0 a=1 b=1 ( 1 t 1 t+1 )dt =logtlog( t+1 ) =log e x log( e x +1 ) =xlog( e x +1 )

    4. e x =t e x = dt dx 1 t t 1 · 1 t dx = 1 t 2 1 dx = 1 ( t+1 )( t1 ) dx a t+1 + b t1 = ( a+b )t+( a+b ) t 2 1 a+b=0 a+b=1 b=a aa=1 a= 1 2 b= 1 2 1 ( t+1 )( t1 ) dx = ( 1 2( t+1 ) + 1 2( t1 ) )dx = 1 2 ( log( t+1 )+log( t1 ) ) = 1 2 ( log( e x +1 )+log| e x 1 | )

    5. 1+x =t 1 2 ( 1+x ) 1 2 = dt dx dx=2 ( 1+x ) 1 2 dt 1+x= t 2 x= t 2 1 t x 2 ( 1+x ) 1 2 dt = t t 2 1 2tdt =2 t 2 t 2 1 dt =2 t 2 1+1 t 2 1 dt =2 ( 1 1 t 2 1 )dt =2t2 1 t 2 1 dt a t+1 + b t1 = ( a+b )ta+b t 2 1 a+b=0 a+b=1 b=a aa=1 a= 1 2 b= 1 2 2t2 ( 1 2( t+1 ) + 1 2( t1 ) )dt =2t+ ( 1 t+1 1 t1 )dt =2t+log( t+1 )log( t1 ) =2 1+x +log 2+x log| x |

    6. x =t 1 2 x 1 2 = dt dx dx=2 x 1 2 dt 1t 1+t 2tdt =2 t 2 +t t+1 dt =2 ( t+2 2 t+1 )dt = t 2 +4t4log( t+1 ) =x+4 x 4log( x +1 )

コード(Emacs)

Python 3

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

from sympy import pprint, symbols, Integral, log, exp, sqrt, plot

print('19.')
x = symbols('x')
fs = [log(1 + x) / x ** 2,
      x * log(x ** 2 + 1),
      1 / (exp(x) + 1),
      1 / (exp(x) - (exp(-x))),
      sqrt(1 + x) / x,
      (1 - sqrt(x)) / (1 + sqrt(x))]

for i, f in enumerate(fs, 1):
    print(f'({i})')
    I = Integral(f, x)
    pprint(I)
    I = I.doit()
    pprint(I)
    for j, g in enumerate([f, I]):
        p = plot(g, show=False, legend=True)
        p.save(f'sample19_{i}_{j}.svg')
    print()

入出力結果(Terminal, IPython)

$ ./sample19.py
19.
(1)
⌠              
⎮ log(x + 1)   
⎮ ────────── dx
⎮      2       
⎮     x        
⌡              
                      log(x + 1)
log(x) - log(x + 1) - ──────────
                          x     

(2)
⌠                 
⎮      ⎛ 2    ⎞   
⎮ x⋅log⎝x  + 1⎠ dx
⌡                 
 2    ⎛ 2    ⎞    2      ⎛ 2    ⎞
x ⋅log⎝x  + 1⎠   x    log⎝x  + 1⎠
────────────── - ── + ───────────
      2          2         2     

(3)
⌠          
⎮   1      
⎮ ────── dx
⎮  x       
⎮ ℯ  + 1   
⌡          
       ⎛ x    ⎞
x - log⎝ℯ  + 1⎠

(4)
⌠            
⎮    1       
⎮ ──────── dx
⎮  x    -x   
⎮ ℯ  - ℯ     
⌡            
   ⎛ x    ⎞      ⎛ x    ⎞
log⎝ℯ  - 1⎠   log⎝ℯ  + 1⎠
─────────── - ───────────
     2             2     

(5)
⌠             
⎮   _______   
⎮ ╲╱ x + 1    
⎮ ───────── dx
⎮     x       
⌡             
⎧    _______          ⎛  _______⎞                 
⎪2⋅╲╱ x + 1  - 2⋅acoth⎝╲╱ x + 1 ⎠  for │x + 1│ > 1
⎨                                                 
⎪    _______          ⎛  _______⎞                 
⎩2⋅╲╱ x + 1  - 2⋅atanh⎝╲╱ x + 1 ⎠     otherwise   

(6)
⌠           
⎮ -√x + 1   
⎮ ─────── dx
⎮  √x + 1   
⌡           
4⋅√x - x - 4⋅log(√x + 1)

$

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="-10">
<label for="x2">x2 = </label>
<input id="x2" type="number" value="10">
<br>
<label for="y1">y1 = </label>
<input id="y1" type="number" value="-10">
<label for="y2">y2 = </label>
<input id="y2" type="number" value="10">

<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="sample19.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 f10 = (x) => Math.log(x + 1) / x **  2,
    f40 = (x) => 1 / (Math.exp(x) - Math.exp(-x)),
    f50 = (x) => Math.sqrt(x + 1) / x;

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 = [[f10, 'red'],
               [f40 , 'green'],
               [f50,'blue']],
        fns1 = [],
        fns2 = [];

    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]);
            }
        }
    });
    fns1.forEach((o) => {
        let [fn, color] = o;
        
        lines.push([x1, fn(x1), x2, fn(x2), color]);
    });
    fns2.forEach((o) => {
        let [fn, color] = o;

        for (let x = x1; x <= x2; x += dx0) {
            let g = fn(x);
            
            lines.push([x1, g(x1), x2, g(x2), 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('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.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.append('g')
        .attr('transform', `translate(0, ${height - padding})`)
        .call(xaxis);

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

    [fns, fns1, fns2].forEach((fs) => p(fs.join('\n')));
};

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







0 コメント:

コメントを投稿