数学 - Python - JavaScript - 細分による加法 - 積分法 – 不定積分の計算 - 部分積分法(対数関数、累乗(べき乗)、漸化式)

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)、部分積分法、問15.を取り組んでみる。

15. 
    $\begin{array}{l}{\displaystyle \int {\left(\log x\right)}^{n}dx}\\ =x{\left(\log x\right)}^n-{\displaystyle \int xn{\left(\log x\right)}^{n-1}\frac{1}{x}dx}\\ =x{\left(\log x\right)}^n-n{\displaystyle \int {\left(\log x\right)}^{n-1}dx}\\ =x{\left(\log x\right)}^n-n{I}_{n-1}\end{array}$

コード(Emacs)

Python 3

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

from sympy import pprint, symbols, Integral, log, solve

print('15.')
x = symbols('x')
n = symbols('n', positive=True, integer=True)
f = Integral(log(x) ** n, x)

pprint(f)

for n0 in range(1, 10):
    eq1 = f.subs({n: n0})
    eq2 = x * log(x) ** n0 - n0 * f.subs({n: n0 - 1})
    pprint(eq1)
    pprint(eq2)
    print(eq1.doit() == eq2.doit())

入出力結果(Terminal, IPython)

$ ./sample15.py
15.
⌠           
⎮    n      
⎮ log (x) dx
⌡           
⌠          
⎮ log(x) dx
⌡          
           ⌠     
x⋅log(x) - ⎮ 1 dx
           ⌡     
True
⌠           
⎮    2      
⎮ log (x) dx
⌡           
     2        ⌠          
x⋅log (x) - 2⋅⎮ log(x) dx
              ⌡          
True
⌠           
⎮    3      
⎮ log (x) dx
⌡           
              ⌠           
     3        ⎮    2      
x⋅log (x) - 3⋅⎮ log (x) dx
              ⌡           
True
⌠           
⎮    4      
⎮ log (x) dx
⌡           
              ⌠           
     4        ⎮    3      
x⋅log (x) - 4⋅⎮ log (x) dx
              ⌡           
True
⌠           
⎮    5      
⎮ log (x) dx
⌡           
              ⌠           
     5        ⎮    4      
x⋅log (x) - 5⋅⎮ log (x) dx
              ⌡           
True
⌠           
⎮    6      
⎮ log (x) dx
⌡           
              ⌠           
     6        ⎮    5      
x⋅log (x) - 6⋅⎮ log (x) dx
              ⌡           
True
⌠           
⎮    7      
⎮ log (x) dx
⌡           
              ⌠           
     7        ⎮    6      
x⋅log (x) - 7⋅⎮ log (x) dx
              ⌡           
True
⌠           
⎮    8      
⎮ log (x) dx
⌡           
              ⌠           
     8        ⎮    7      
x⋅log (x) - 8⋅⎮ log (x) dx
              ⌡           
True
⌠           
⎮    9      
⎮ log (x) dx
⌡           
              ⌠           
     9        ⎮    8      
x⋅log (x) - 9⋅⎮ log (x) dx
              ⌡           
True
$

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="sample15.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 f = (n) => (x) => Math.log(x) ** n;

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 = ['red','green', 'blue', 'orange', 'brown', 'purple']
        .map((color, i) => [f(i), color]),
        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 =