数学 - Python - JavaScript - 解析学 - 積分 - 積分の計算 - 部分分数(置換積分法、対数関数、逆正接関数)

解析入門 原書第3版
原著

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)
• Nebo(Windows アプリ)
• iPad Pro + Apple Pencil
• MyScript Nebo(iPad アプリ)
• 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

解析入門 原書第3版 (S.ラング(著)、松坂 和夫(翻訳)、片山 孝次(翻訳)、岩波書店)の第3部(積分)、第11章(積分の計算)、4(部分分数)、練習問題12.を取り組んでみる。

12. 
    
    $\begin{array}{}\frac{a}{x}+\frac{bx+c}{x^2+x+1}\\ =\frac{\left(a+b\right)x^2+\left(a+c\right)x+a}{x\left(x^2+x+1\right)}\\ a+b=0\\ a+c=0\\ a=1\\ b=-1\\ c=-1\\ \int \; <!--\; integral\; -->\frac{1}{x}\mathrm{dx}-\int \; <!--\; integral\; -->\frac{x+1}{x^2+x+1}\mathrm{dx}\\ \frac{x+1}{x^2+x+1}\\ =\frac{x+\frac{1}{2}+\frac{1}{2}}{{\left(x+\frac{1}{2}\right)}^2+\frac{3}{4}}\\ =\frac{x+\frac{1}{2}}{{\left(x+\frac{1}{2}\right)}^2+\frac{3}{4}}+\frac{\frac{1}{2}}{{\left(x+\frac{1}{2}\right)}^2+\frac{3}{4}}\\ x+\frac{1}{2}=t\\ \frac{\mathrm{dx}}{\mathrm{dt}}=1\\ \int \; <!--\; integral\; -->\frac{t}{t^2+\frac{3}{4}}\mathrm{dt}+\frac{1}{2}\int \; <!--\; integral\; -->\frac{1}{t^2+\frac{3}{4}}\mathrm{dt}\\ t=\frac{\sqrt{3}}{2}u\\ \frac{\mathrm{dt}}{du}=\frac{\sqrt{3}}{2}\\ \int \; <!--\; integral\; -->\frac{u}{u^2+1}du+\frac{\sqrt{3}}{4}·\; <!--\; middle\; dot\; -->\frac{4}{3}\int \; <!--\; integral\; -->\frac{1}{u^2+1}du\end{array}$

    よって、求める積分は

    $\begin{array}{}\log x-\frac{1}{2}\log \left(u^2+1\right)-\frac{1}{\sqrt{3}}\arctan u\\ =\log x-\frac{1}{2}\log \left(\frac{4}{3}{t}^2+1\right)-\frac{1}{\sqrt{3}}\arctan \left(\frac{2}{\sqrt{3}}t\right)\\ =\log x-\frac{1}{2}\log \left(\frac{4}{3}{\left(x+\frac{1}{2}\right)}^2+1\right)-\frac{1}{\sqrt{3}}\arctan \left(\frac{2}{\sqrt{3}}{\left(x+\frac{1}{2}\right)}^2\right)\end{array}$

コード(Emacs)

Python 3

#!/usr/bin/env python3
from sympy import pprint, symbols, Integral, plot, atan, sqrt, log, Rational, Derivative

print('12.')
x = symbols('x')
f = 1 / (x * (x ** 2 + x + 1))
I = Integral(f, x)

for t in [I, I.doit()]:
    pprint(t)
    print()

p = plot(x, x ** 2 + x + 1, x * (x ** 2 + x + 1),
         f, ylim=(-5, 5), legend=True, show=False)
colors = ['red', 'green', 'blue', 'brown']
for i, color in enumerate(colors):
    p[i].line_color = color
p.save('sample13.svg')

expr = log((Rational(4, 3) * (x + Rational(1, 2)) ** 2 + 1).factor())
pprint(expr)
pprint((log(x ** 2 + x + 1) - expr).expand())

入出力結果(Terminal, Jupyter(IPython))

$ ./sample13.py
12.
⌠                  
⎮       1          
⎮ ────────────── dx
⎮   ⎛ 2        ⎞   
⎮ x⋅⎝x  + x + 1⎠   
⌡                  

                                  ⎛2⋅√3⋅x   √3⎞
            ⎛ 2        ⎞   √3⋅atan⎜────── + ──⎟
         log⎝x  + x + 1⎠          ⎝  3      3 ⎠
log(x) - ─────────────── - ────────────────────
                2                   3          

   ⎛  ⎛ 2        ⎞⎞
   ⎜4⋅⎝x  + x + 1⎠⎟
log⎜──────────────⎟
   ⎝      3       ⎠
-2⋅log(2) + log(3)
$

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="0.001" 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="sample13.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'),
    input_n0 = document.querySelector('#n0'),
    inputs = [input_r, input_dx, input_x1, input_x2, input_y1, input_y2],
    p = (x) => pre0.textContent += x + '\n';

let f = (x) => x,
    g = (x) => x ** 2 + x + 1,
    h = (x) => f(x) * g(x),
    fns = [[f, 'red'],
           [g, 'green'],
           [h, 'blue'],
           [(x) => 1 / h(x), 'brown']];

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
        .forEach((o) => {
            let [f, color] = o;
            for (let x = x1; x <= x2; x += dx) {
                let y = f(x);

                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('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].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 =