数学 - Python - JavaScript - 解析学 - 積分法 - 不定積分の計算(有理式、置換法、三角関数、正弦、余弦、正接)

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)、8.1(不定積分の計算)、問題7.を取り組んでみる。

7. 
   $\begin{array}{l}\frac{2\tan \frac{x}{2}}{1+{\tan }^2\frac{x}{2}}\\ =\frac{2\frac{\sin \frac{x}{2}}{\cos \frac{x}{2}}}{1+\frac{{\sin }^2\frac{x}{2}}{{\cos }^2\frac{x}{2}}}\\ =2\sin \frac{x}{2}\cos \frac{x}{2}\\ =\sin x\\ \\ \frac{1-{\tan }^2\frac{x}{2}}{1+{\tan }^2\frac{x}{2}}\\ =\frac{1-\frac{{\sin }^2\frac{x}{2}}{{\cos }^2\frac{x}{2}}}{1+\frac{{\sin }^2\frac{x}{2}}{{\cos }^2\frac{x}{2}}}\\ ={\cos }^2\frac{x}{2}-{\sin }^2\frac{x}{2}\\ =\cos x\\ \\ \frac{dx}{dt}=\frac{1}{\frac{dt}{dx}}\\ =\frac{1}{\frac{1}{{\cos }^2\frac{x}{2}}·\frac{1}{2}}\\ =\frac{2}{\frac{{\sin }^2\frac{x}{2}+{\cos }^2\frac{x}{2}}{{\cos }^2\frac{x}{2}}}\\ =\frac{2}{1+{\tan }^2\frac{x}{2}}\\ =\frac{2}{1+t^2}\end{array}$
   1. 
      $\begin{array}{l}{\displaystyle \int \frac{1}{1+\frac{1-t^2}{1+t^2}}·\frac{2}{1+t^2}dt}\\ ={\displaystyle \int \frac{2}{1+t^2+1-t^2}·dt}\\ ={\displaystyle \int 1dt}\\ =t\\ =\tan \frac{x}{2}\end{array}$
   2. 
      $\begin{array}{l}{\displaystyle \int \frac{1}{\frac{1-t^2}{1+t^2}}·\frac{2}{1+t^2}dt}\\ ={\displaystyle \int \frac{2}{1-t^2}dt}\\ ={\displaystyle \int \frac{2}{\left(1+t\right)\left(1-t\right)}dt}\\ \frac{A}{1+t}+\frac{B}{1-t}\\ =\frac{\left(B-A\right)t+A+B}{\left(1+t\right)\left(1-t\right)}\\ B-A=0\\ A+B=2\\ A=B\\ 2A=2\\ A=1\\ B=1\\ {\displaystyle \int \frac{2}{\left(1+t\right)\left(1-t\right)}dt}\\ ={\displaystyle \int \frac{1}{1+t}dt}+{\displaystyle \int \frac{1}{1-t}dt}\\ =\log \left|1+t\right|-\log \left|1-t\right|\\ =\log \left|\frac{1+t}{1-t}\right|\\ =\log \left|\frac{1+\tan \frac{x}{2}}{1-\tan \frac{x}{2}}\right|\end{array}$
   3. 
      $\begin{array}{l}{\displaystyle \int \frac{\frac{2t}{1+t^2}}{1+\frac{2t}{1+t^2}}·\frac{2}{1+t^2}dt}\\ ={\displaystyle \int \frac{2t}{1+t^2+2t}·\frac{2}{1+t^2}dt}\\ =4{\displaystyle \int \frac{t}{{\left(t+1\right)}^2\left(t^2+1\right)}dt}\\ \frac{A}{t+1}+\frac{B}{{\left(t+1\right)}^2}+\frac{Ct+D}{t^2+1}\\ =\frac{A\left(t+1\right)\left(t^2+1\right)+B\left(t^2+1\right)+\left(Ct+D\right){\left(t+1\right)}^2}{{\left(t+1\right)}^2\left(t^2+1\right)}\\ A\left(t+1\right)\left(t^2+1\right)+B\left(t^2+1\right)+\left(Ct+D\right){\left(t+1\right)}^2\\ =A\left(t^3+t^2+t+1\right)+B{t}^2+B+\left(Ct+D\right)\left(t^2+2t+1\right)\\ =\left(A+C\right)t^3+\left(A+B+2C+D\right)t^2+\left(A+C+2D\right)t+A+B+D\\ A+C=0\\ A+B+2C+D=0\\ A+C+2D=1\\ A+B+D=0\\ \\ C=-A\\ A+B-2A+D=0\\ A-A+2D=1\\ A+B+D=0\\ \\ D=-A-B\\ A+B-2A-A-B=0\\ A=0\\ D=-B\\ 2D=1\\ D=\frac{1}{2}\\ B=-\frac{1}{2}\\ C=0\\ \\ 4{\displaystyle \int \frac{t}{{\left(t+1\right)}^2\left(t^2+1\right)}dt}\\ =4{\displaystyle \int \left(\frac{-\frac{1}{2}}{{\left(t+1\right)}^2}+\frac{\frac{1}{2}}{t^2+1}\right)dt}\\ =2{\displaystyle \int \left(\frac{-1}{{\left(t+1\right)}^2}+\frac{1}{t^2+1}\right)dt}\\ =2\left({\left(t+1\right)}^{-1}+\arctan t\right)\\ =2\left(\frac{1}{\tan \frac{x}{2}+1}+\arctan \left(\tan \frac{x}{2}\right)\right)\\ =\frac{2}{\tan \frac{x}{2}+1}+2\frac{x}{2}\\ =\frac{2}{\tan \frac{x}{2}+1}+x\end{array}$

コード(Emacs)

Python 3

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

from sympy import pprint, symbols, Integral, sin, cos

print('7.')
x = symbols('x')

fs = [1 / (1 + cos(x)),
      1 / cos(x),
      sin(x) / (1 + sin(x))]


for i, f in enumerate(fs, 1):
    print(f'({i})')
    I = Integral(f, x)
    pprint(I)
    I = I.doit()
    pprint(I)
    print('factor:')
    pprint(I.factor())
    print('expand:')
    pprint(I.expand())

入出力結果(Terminal, IPython)

$ ./sample7.py
7.
(1)
⌠              
⎮     1        
⎮ ────────── dx
⎮ cos(x) + 1   
⌡              
   ⎛x⎞
tan⎜─⎟
   ⎝2⎠
factor:
   ⎛x⎞
tan⎜─⎟
   ⎝2⎠
expand:
   ⎛x⎞
tan⎜─⎟
   ⎝2⎠
(2)
⌠          
⎮   1      
⎮ ────── dx
⎮ cos(x)   
⌡          
  log(sin(x) - 1)   log(sin(x) + 1)
- ─────────────── + ───────────────
         2                 2       
factor:
-log(sin(x) - 1) + log(sin(x) + 1)
──────────────────────────────────
                2                 
expand:
  log(sin(x) - 1)   log(sin(x) + 1)
- ─────────────── + ───────────────
         2                 2       
(3)
⌠              
⎮   sin(x)     
⎮ ────────── dx
⎮ sin(x) + 1   
⌡              
      ⎛x⎞                           
 x⋅tan⎜─⎟                           
      ⎝2⎠        x            2     
────────── + ────────── + ──────────
   ⎛x⎞          ⎛x⎞          ⎛x⎞    
tan⎜─⎟ + 1   tan⎜─⎟ + 1   tan⎜─⎟ + 1
   ⎝2⎠          ⎝2⎠          ⎝2⎠    
factor:
     ⎛x⎞        
x⋅tan⎜─⎟ + x + 2
     ⎝2⎠        
────────────────
      ⎛x⎞       
   tan⎜─⎟ + 1   
      ⎝2⎠       
expand:
      ⎛x⎞                           
 x⋅tan⎜─⎟                           
      ⎝2⎠        x            2     
────────── + ────────── + ──────────
   ⎛x⎞          ⎛x⎞          ⎛x⎞    
tan⎜─⎟ + 1   tan⎜─⎟ + 1   tan⎜─⎟ + 1
   ⎝2⎠          ⎝2⎠          ⎝2⎠
$   

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="-10">
<label for="x2">x2 = </label>
<input id="x2" type="number" value="10">
<br>
<label for="y1">y1 = </label>
<input id="y1" type="number" value="-10">
<label for="y2">y2 = </label>
<input id="y2" type="number" value="10">

<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="sample7.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 f1 = (x) => 1 / (1 + Math.cos(x)),
    f2 = (x) => 1 / Math.cos(x),
    f3 = (x) => Math.sin(x) / (1 + Math.sin(x));

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 = [[f1, 'red'],
               [f2, 'green'],
               [f3, 'blue']];

    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 =