数学 - Python - JavaScript - 細分による加法 - 積分法 – 定積分の性質と計算 - 定積分の置換積分法(三角関数(正弦、余弦)、指数関数、累乗(べき乗))

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)、19.3(定積分の性質と計算)、定積分の置換積分法、問26.を取り組んでみる。

26. 1. 
       $\begin{array}{l}2x-1=t\\ 2=\frac{dt}{dx}\\ \\ x=1\\ t=2-1=1\\ x=3\\ t=6-1=5\\ \\ {\displaystyle {\int }_1^5{t}^3\frac{1}{2}dt}\\ =\frac{1}{2}{\left[\frac{1}{4}{t}^4\right]}_1^5\\ =\frac{1}{8}\left(5^4-1^4\right)\\ =\frac{1}{8}\left(625-1\right)\\ =\frac{624}{8}\\ =78\end{array}$
    2. 
       $\begin{array}{l}\cos x=t\\ -\sin x=\frac{dt}{dx}\\ \\ x=0\\ t=1\\ x=\frac{\pi }{2}\\ t=\cos \frac{\pi }{2}=0\\ \\ -{\displaystyle {\int }_1^0{t}^{2}dt}\\ =-{\left[\frac{1}{3}{t}^3\right]}_1^0\\ =\frac{1}{3}\end{array}$
    3. 
       $\begin{array}{l}\cos x=t\\ -\sin x=\frac{dt}{dx}\\ \\ x=0\\ t=\cos 0=1\\ x=\pi \\ t=\cos \pi =-1\\ \\ -{\displaystyle {\int }_1^{-1}{t}^{3}dt}\\ ={\displaystyle {\int }_{-1}^1{t}^{3}dt}\\ =0\end{array}$
    4. 
       $\begin{array}{l}\sin x=t\\ \cos x=\frac{dt}{dx}\\ \\ \sin 0=0\\ \sin \frac{\pi }{6}=\frac{1}{2}\\ \\ {\displaystyle {\int }_0^{\frac{1}{2}}{t}^{4}dt}\\ ={\left[\frac{1}{5}{t}^5\right]}_0^{\frac{1}{2}}\\ =\frac{1}{5}·\frac{1}{2^5}\\ =\frac{1}{160}\end{array}$
    5. 
       $\begin{array}{l}{x}^2=t\\ 2x=\frac{dt}{dx}\\ \\ t=0^2=0\\ t=1^2=1\\ \\ {\displaystyle {\int }_0^1{e}^t\frac{1}{2}dt}\\ =\frac{1}{2}{\left[e^t\right]}_0^1\\ =\frac{1}{2}\left(e-1\right)\end{array}$
    6. 
       $\begin{array}{l}{x}^2=t\\ 2x=\frac{dt}{dx}\\ \\ t=0^2=0\\ t={\sqrt{\pi }}^2=\pi \\ \\ {\displaystyle {\int }_0^{\pi }\frac{1}{2}\sin tdt}\\ =\frac{1}{2}{\left[-\cos t\right]}_0^{\pi }\\ =-\frac{1}{2}\left(-1-1\right)\\ =1\end{array}$

コード(Emacs)

Python 3

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

from sympy import pprint, symbols, Integral, plot, sqrt, sin, cos, exp, pi

print('26.')
x = symbols('x')
fs = [((2 * x - 1) ** 3, (1, 3)),
      (sin(x) * cos(x) ** 2, (0, pi / 2)),
      (sin(x) * cos(x) ** 3, (0, pi)),
      (sin(x) ** 4 * cos(x), (0, pi / 6)),
      (x * exp(x ** 2), (0, 1)),
      (x * sin(x**2), (0, sqrt(pi)))]


for i, (f, (x1, x2)) in enumerate(fs, 1):
    try:
        print(f'({i})')
        I = Integral(f, (x, x1, x2))
        for g in [I, I.doit()]:
            pprint(g)
        p = plot(f, show=False, legend=True)
        p.save(f'sample26_{i}.svg')
    except Exception as err:
        print(type(err), err)
    print()

入出力結果(Terminal, IPython)

$ ./sample26.py
26.
(1)
3              
⌠              
⎮          3   
⎮ (2⋅x - 1)  dx
⌡              
1              
78

(2)
π                  
─                  
2                  
⌠                  
⎮           2      
⎮ sin(x)⋅cos (x) dx
⌡                  
0                  
1/3

(3)
π                  
⌠                  
⎮           3      
⎮ sin(x)⋅cos (x) dx
⌡                  
0                  
0

(4)
π                  
─                  
6                  
⌠                  
⎮    4             
⎮ sin (x)⋅cos(x) dx
⌡                  
0                  
1/160

(5)
1           
⌠           
⎮    ⎛ 2⎞   
⎮    ⎝x ⎠   
⎮ x⋅ℯ     dx
⌡           
0           
  1   ℯ
- ─ + ─
  2   2

(6)
√π             
⌠              
⎮       ⎛ 2⎞   
⎮  x⋅sin⎝x ⎠ dx
⌡              
0              
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="-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="sample26.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 = (x) => x * Math.exp(x ** 2);

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 = [],
        x3 = Math.sqrt(3),
        lines = [[0, y1, 0, y2, 'red'],
                 [1, y1, 1, y2, 'red']],
        fns = [[f, 'green']],
        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();

(x) => x * Math.exp(x ** 2),green

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