数学 - Python - JavaScript - 解析学 - 積分法 - 定積分の計算(置換積分法、部分積分法、対数関数、累乗関数、三角関数(逆正弦))

解析入門〈2〉

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

解析入門〈2〉(松坂 和夫(著)、岩波書店)の第8章(積分の計算)、8.2(定積分の計算)、問題5-1、2.を取り組んでみる。

5. 1. 
      $\begin{array}{l}x=\frac{a+b}{2}+t\\ x=a,t=\frac{a-b}{2}\\ x=b,t=\frac{b-a}{2}\\ {\displaystyle {\int }_{\frac{a-b}{2}}^{\frac{b-a}{2}}\frac{dt}{\sqrt{\left(\left(\frac{a+b}{2}+t\right)-a\right)\left(b-\left(\frac{a+b}{2}+t\right)\right)}}}\\ ={\displaystyle {\int }_{\frac{a-b}{2}}^{\frac{b-a}{2}}\frac{dt}{\sqrt{\left(\frac{b-a}{2}+t\right)\left(\frac{b-a}{2}-t\right)}}}\\ \frac{b-a}{2}=h\\ {\displaystyle {\int }_{\frac{a-b}{2}}^{\frac{b-a}{2}}\frac{dt}{\sqrt{\left(h+t\right)\left(h-t\right)}}}\\ ={\displaystyle {\int }_{\frac{a-b}{2}}^{\frac{b-a}{2}}\frac{dt}{\sqrt{h^2-t^2}}}\\ ={\left[\arcsin \frac{t}{h}\right]}_{\frac{a-b}{2}}^{\frac{b-a}{2}}\\ =\arcsin \frac{\frac{b-a}{2}}{h}-\arcsin \frac{\frac{a-b}{2}}{h}\\ =\arcsin \frac{\frac{b-a}{2}}{\frac{b-a}{2}}-\arcsin \frac{\frac{a-b}{2}}{\frac{b-a}{2}}\\ =\arcsin 1-\arcsin \left(-1\right)\\ =\frac{\pi }{2}-\left(-\frac{\pi }{2}\right)\\ =\pi \end{array}$
   2. 
      $\begin{array}{l}{\displaystyle \int x^{-\alpha }\log xdx}\\ =\frac{x^{1-\alpha }\log x}{1-\alpha }-{\displaystyle \int \frac{x^{-\alpha }}{1-\alpha }dx}\\ =\frac{1}{1-\alpha }\left(x^{1-\alpha }\log x-{\displaystyle \int x^{-\alpha }dx}\right)\\ =\frac{1}{1-\alpha }\left(x^{1-\alpha }\log x-\frac{x^{1-\alpha }}{1-\alpha }\right)\\ =\frac{x^{1-\alpha }}{{\left(1-\alpha \right)}^2}\left(\left(1-\alpha \right)\log x-1\right)\\ \\ {\displaystyle {\int }_0^1{x}^{-\alpha }\log xdx}\\ ={\left[\frac{x^{1-\alpha }}{{\left(1-\alpha \right)}^2}\left(\left(1-\alpha \right)\log x-1\right)\right]}_0^1\\ =-\frac{1}{{\left(1-\alpha \right)}^2}\end{array}$

コード(Emacs)

Python 3

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

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

print('5.')
x, a, b = symbols('x a b')
fs = [(1 / sqrt((x - a) * (b - x)), (a, b)),
      (log(x) / x ** a, (0, 1))]

for i, (f, (x1, x2)) in enumerate(fs, 1):
    I = Integral(f, (x, a, b))
    pprint(I)
    pprint(I.doit())
    try:
        p = plot(f, show=False, legend=True)
        p.save(f'sample5_{i}.svg')
    except Exception as err:
        print(type(err), err)

入出力結果(Terminal, IPython)

$ ./sample5.py
5.
b                        
⌠                        
⎮          1             
⎮ ──────────────────── dx
⎮   __________________   
⎮ ╲╱ (-a + x)⋅(b - x)    
⌡                        
a                        
b                        
⌠                        
⎮          1             
⎮ ──────────────────── dx
⎮   __________________   
⎮ ╲╱ (-a + x)⋅(b - x)    
⌡                        
a                        
<class 'ValueError'> The same variable should be used in all univariate expressions being plotted.
b              
⌠              
⎮  -a          
⎮ x  ⋅log(x) dx
⌡              
a              
b              
⌠              
⎮  -a          
⎮ x  ⋅log(x) dx
⌡              
a              
<class 'ValueError'> The same variable should be used in all univariate expressions being plotted.
$

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">
<br>
<label for="a0">a = </label>
<input id="a0" type="number" value="-1">
<label for="b0">b = </label>
<input id="b0" type="number" value="2">

<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="sample5.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_a0 = document.querySelector('#a0'),
    input_b0 = document.querySelector('#b0'),
    inputs = [input_r, input_dx, input_x1, input_x2, input_y1, input_y2,
              input_a0, input_b0],
    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 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),
        a0 = parseFloat(input_a0.value),
        b0 = parseFloat(input_b0.value);
        
    
    if (r === 0 || dx === 0 || x1 > x2 || y1 > y2) {
        return;
    }

    let points = [],
        f = (x) => 1 / Math.sqrt((x - a0) * (x - b0)),
        g = (x) => Math.log(x) / x ** a0,
        lines = [[a0, y1, a0, y2, 'red'],
                 [b0, y1, b0, y2, 'red'],
                 [0, y1, 0, y2, 'blue'],
                 [1, y1, 1, y2, 'blue']],
        fns = [[f, 'green'],
               [g, 'orange']];

    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]);
                }
            }
        });
    
    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('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.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.append('g')
        .attr('transform', `translate(0, ${height - padding})`)
        .call(xaxis);

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

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

r = dx =
x1 = x2 =
y1 = y2 =
a = b =