数学 - Python - JavaScript - 細分による加法 - 積分法 – 不定積分の計算 - 分数関数の積分(指数関数、対数関数、部分積分法・置換積分法、分数関数の積分法)

数学読本〈5〉

学習環境

• Surface 3 (4G LTE)、Surface 3 タイプ カバー、Surface ペン(端末)
• Windows 10 Pro (OS)
• 数式入力ソフト(TeX, MathML): MathType
• MathML対応ブラウザ: Firefox、Safari
• MathML非対応ブラウザ(Internet Explorer, Microsoft Edge, Google Chrome...)用JavaScript Library: MathJax
• 参考書籍
  • Pythonからはじめる数学入門
  • JavaScriptリファレンス 第6版
  • インタラクティブ・データビジュアライゼーション

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。

19. 1. 
       $\begin{array}{l}{\displaystyle \int \frac{\log \left(1+x\right)}{x^2}dx}\\ =-\frac{1}{x}\log \left(1+x\right)-{\displaystyle \int -\frac{1}{x\left(1+x\right)}dx}\\ =-\frac{\log \left(1+x\right)}{x}+{\displaystyle \int \frac{1}{x\left(x+1\right)}dx}\\ \frac{a}{x}+\frac{b}{x+1}\\ =\frac{\left(a+b\right)x+a}{x\left(x+1\right)}\\ a+b=0\\ a=1\\ b=-1\\ -\frac{\log \left(1+x\right)}{x}+{\displaystyle \int \left(\frac{1}{x}-\frac{1}{x+1}\right)dx}\\ =-\frac{\log \left(1+x\right)}{x}+\log \left|x\right|-\log \left(x+1\right)\end{array}$
    2. 
       $\begin{array}{l}{\displaystyle \int x\log \left(x^2+1\right)dx}\\ =\frac{1}{2}{x}^2\log \left(x^2+1\right)-{\displaystyle \int \frac{2x^2}{x^2+1}dx}\\ =\frac{x^2\log \left(x^2+1\right)}{2}-{\displaystyle \int \frac{2\left(x^2+1\right)-2}{x^2+1}dx}\\ =\frac{x^2\log \left(x^2+1\right)}{2}-\left({\displaystyle \int 2dx-2{\displaystyle \int \frac{1}{x^2+1}dx}}\right)\\ =\frac{x^2\log \left(x^2+1\right)}{2}-2x+2\arctan x\end{array}$
    3. 
       $\begin{array}{l}{e}^x=t\\ e^x=\frac{dt}{dx}\\ {\displaystyle \int \frac{1}{t+1}\frac{1}{t}dt}\\ ={\displaystyle \int \frac{1}{t\left(t+1\right)}dt}\\ \frac{a}{t}+\frac{b}{t+1}\\ =\frac{\left(a+b\right)t+a}{t\left(t+1\right)}\\ a+b=0\\ a=1\\ b=-1\\ {\displaystyle \int \left(\frac{1}{t}-\frac{1}{t+1}\right)dt}\\ =\log t-\log \left(t+1\right)\\ =\log e^x-\log \left(e^x+1\right)\\ =x-\log \left(e^x+1\right)\end{array}$
    4. 
       $\begin{array}{l}{e}^x=t\\ e^x=\frac{dt}{dx}\\ {\displaystyle \int \frac{1}{t-t^{-1}}·\frac{1}{t}dx}\\ ={\displaystyle \int \frac{1}{t^2-1}dx}\\ ={\displaystyle \int \frac{1}{\left(t+1\right)\left(t-1\right)}dx}\\ \frac{a}{t+1}+\frac{b}{t-1}\\ =\frac{\left(a+b\right)t+\left(-a+b\right)}{t^2-1}\\ a+b=0\\ -a+b=1\\ b=-a\\ -a-a=1\\ a=-\frac{1}{2}\\ b=\frac{1}{2}\\ {\displaystyle \int \frac{1}{\left(t+1\right)\left(t-1\right)}dx}\\ ={\displaystyle \int \left(-\frac{1}{2\left(t+1\right)}+\frac{1}{2\left(t-1\right)}\right)dx}\\ =\frac{1}{2}\left(-\log \left(t+1\right)+\log \left(t-1\right)\right)\\ =\frac{1}{2}\left(-\log \left(e^x+1\right)+\log \left|e^x-1\right|\right)\end{array}$
    5. 
       $\begin{array}{l}\sqrt{1+x}=t\\ \frac{1}{2}{\left(1+x\right)}^{-\frac{1}{2}}=\frac{dt}{dx}\\ dx=2{\left(1+x\right)}^{\frac{1}{2}}dt\\ 1+x=t^2\\ x=t^2-1\\ {\displaystyle \int \frac{t}{x}2{\left(1+x\right)}^{\frac{1}{2}}dt}\\ ={\displaystyle \int \frac{t}{t^2-1}2tdt}\\ =2{\displaystyle \int \frac{t^2}{t^2-1}dt}\\ =2{\displaystyle \int \frac{t^2-1+1}{t^2-1}dt}\\ =2{\displaystyle \int \left(1-\frac{1}{t^2-1}\right)dt}\\ =2t-2{\displaystyle \int \frac{1}{t^2-1}dt}\\ \frac{a}{t+1}+\frac{b}{t-1}\\ =\frac{\left(a+b\right)t-a+b}{t^2-1}\\ a+b=0\\ -a+b=1\\ b=-a\\ -a-a=1\\ a=-\frac{1}{2}\\ b=\frac{1}{2}\\ 2t-2{\displaystyle \int \left(-\frac{1}{2\left(t+1\right)}+\frac{1}{2\left(t-1\right)}\right)dt}\\ =2t+{\displaystyle \int \left(\frac{1}{t+1}-\frac{1}{t-1}\right)dt}\\ =2t+\log \left(t+1\right)-\log \left(t-1\right)\\ =2\sqrt{1+x}+\log \sqrt{2+x}-\log \left|x\right|\end{array}$
    6. 
       $\begin{array}{l}\sqrt{x}=t\\ \frac{1}{2}{x}^{-\frac{1}{2}}=\frac{dt}{dx}\\ dx=2x^{\frac{1}{2}}dt\\ {\displaystyle \int \frac{1-t}{1+t}2tdt}\\ =2{\displaystyle \int \frac{-t^2+t}{t+1}dt}\\ =2{\displaystyle \int \left(-t+2-\frac{2}{t+1}\right)dt}\\ =-t^2+4t-4\log \left(t+1\right)\\ =-x+4\sqrt{x}-4\log \left(\sqrt{x}+1\right)\end{array}$

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();

r = dx =
x1 = x2 =
y1 = y2 =