数学 - Python - JavaScript - 解析学 - 積分 - 積分の計算 - 部分積分法(累乗(べき乗)、平方根)

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)、2(部分積分法)、練習問題14.を取り組んでみる。

14. 
    
    $\begin{array}{}\int \; <!--\; integral\; -->x^5\sqrt{1-x^2}\mathrm{dx}\\ =\int \; <!--\; integral\; -->\left(-\frac{1}{3}{x}^4\right)\left(-3x\sqrt{1-x^2}\right)\mathrm{dx}\\ =-\frac{1}{3}{x}^4{\left(1-x^2\right)}^{\left(\frac{3}{2}\right)}+\frac{4}{3}\int \; <!--\; integral\; -->x^3{\left(1-x^2\right)}^{\left(\frac{3}{2}\right)}\mathrm{dx}\\ =-\frac{1}{3}{x}^4{\left(1-x^2\right)}^{\left(\frac{3}{2}\right)}+\frac{4}{3}\int \; <!--\; integral\; -->\left(-\frac{1}{5}{x}^2\right)\left(-5x{\left(1-x^2\right)}^{\left(\frac{3}{2}\right)}\right)\mathrm{dx}\\ =-\frac{1}{3}{x}^4{\left(1-x^2\right)}^{\left(\frac{3}{2}\right)}+\frac{4}{3}\left(-\frac{1}{5}{x}^2{\left(1-x^2\right)}^{\left(\frac{5}{2}\right)}+\frac{2}{5}\int \; <!--\; integral\; -->x{\left(1-x^2\right)}^{\left(\frac{5}{2}\right)}\mathrm{dx}\right)\\ =-\frac{1}{3}{x}^4{\left(1-x^2\right)}^{\left(\frac{3}{2}\right)}-\frac{4}{15}{x}^2{\left(1-x^2\right)}^{\left(\frac{5}{2}\right)}+\frac{8}{15}\int \; <!--\; integral\; -->\left(-\frac{1}{7}\right)\left(-7x{\left(1-x^2\right)}^{\left(\frac{5}{2}\right)}\right)\mathrm{dx}\end{array}$

    よって、

    $-\frac{1}{3}{x}^4{\left(1-x^2\right)}^{\left(\frac{3}{2}\right)}-\frac{4}{15}{x}^2{\left(1-x^2\right)}^{\left(\frac{5}{2}\right)}-\frac{8}{105}{\left(1-x^2\right)}^{\left(\frac{7}{2}\right)}$

コード(Emacs)

Python 3

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

x = symbols('x')
f = x ** 5 * sqrt(1 - x ** 2)
I = Integral(f, x)
for t in [I, I.doit()]:
    pprint(t)
    print()

p = plot(x**5, sqrt(1 - x ** 2), f, (x, -1, 1), legend=True, show=False)

for i, color in enumerate(['red', 'green', 'blue']):
    p[i].line_color = color

p.save('sample12.svg')

入出力結果(Terminal, Jupyter(IPython))

$ ./sample12.py
⌠                    
⎮       __________   
⎮  5   ╱    2        
⎮ x ⋅╲╱  - x  + 1  dx
⌡                    

      __________         __________           __________        __________
 6   ╱    2         4   ╱    2           2   ╱    2            ╱    2     
x ⋅╲╱  - x  + 1    x ⋅╲╱  - x  + 1    4⋅x ⋅╲╱  - x  + 1    8⋅╲╱  - x  + 1 
──────────────── - ──────────────── - ────────────────── - ───────────────
       7                  35                 105                 105      

$

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="0.005">
<br>
<label for="x1">x1 = </label>
<input id="x1" type="number" value="-1.5">
<label for="x2">x2 = </label>
<input id="x2" type="number" value="1.5">
<br>
<label for="y1">y1 = </label>
<input id="y1" type="number" value="-1.5">
<label for="y2">y2 = </label>
<input id="y2" type="number" value="1.5">

<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="sample12.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 fns = [[(x) => x ** 5, 'red'],
           [(x) => Math.sqrt(1 - x ** 2), 'green'],
           [(x) => x ** 5 * Math.sqrt(1 - x ** 2), 'blue']];

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 = [],
        fns1 = [],
        fns2 = [];

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

                points.push([x, y, color]);
            }
        });

    fns1
        .forEach((o) => {
            let [f, color] = o;
            
            lines.push([x1, f(x1), x2, f(x2), color]);
        });
    
    fns2
        .forEach((o) => {
            let [f, color] = o;
            for (let x = x1; x <= x2; x += dx0) {
                let g = f(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 =