数学 - Python - JavaScript - 解析学 - 積分法 - 不定積分の計算(指数関数、対数関数、部分積分法、置換積分法、有理関数)

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8.を取り組んでみる。

8. 1. 
      $\begin{array}{l}{\displaystyle \int e^{-x}{\cos }^{2}xdx}\\ ={\displaystyle \int \frac{d}{dx}\left(-e^{-x}\right){\cos }^{2}xdx}\\ =-e^{-x}{\cos }^{2}x-{\displaystyle \int e^{-x}2\cos x\sin xdx}\\ =-e^{-x}{\cos }^{2}x-{\displaystyle \int e^{-x}\sin 2xdx}\\ =-e^{-x}{\cos }^{2}x-{\displaystyle \int \frac{d}{dx}\left(-e^{-x}\right)\sin 2xdx}\\ =-e^{-x}{\cos }^{2}x-\left(-e^{-x}\sin 2x+2{\displaystyle \int e^{-x}\cos 2xdx}\right)\\ =-e^{-x}{\cos }^{2}x+e^{-x}\sin 2x-2{\displaystyle \int e^{-x}\left({\cos }^{2}x-{\sin }^{2}x\right)dx}\\ =-e^{-x}{\cos }^{2}x+e^{-x}\sin 2x-2{\displaystyle \int e^{-x}\left(2{\cos }^{2}x-1\right)dx}\\ =-e^{-x}{\cos }^{2}x+e^{-x}\sin 2x-2\left(2{\displaystyle \int e^{-x}{\cos }^{2}xdx-{\displaystyle \int e^{-x}dx}}\right)\\ =-e^{-x}{\cos }^{2}x+e^{-x}\sin 2x-4{\displaystyle \int e^{-x}{\cos }^{2}xdx-2e^{-x}}\\ 5{\displaystyle \int e^{-x}{\cos }^{2}xdx}=-e^{-x}{\cos }^{2}x+e^{-x}\sin 2x-2e^{-x}\\ {\displaystyle \int e^{-x}{\cos }^{2}xdx}=-\frac{1}{5}{e}^{-x}{\cos }^{2}x+\frac{1}{5}{e}^{-x}\sin 2x-\frac{2}{5}{e}^{-x}\end{array}$
   2. 
      $\begin{array}{l}x=\tan t\\ \frac{dx}{dt}=\frac{1}{{\cos }^{2}t}\\ x=\frac{\sin t}{\cos t}\\ x^2=\frac{{\sin }^{2}t}{{\cos }^{2}t}\\ x^2=\frac{{\sin }^{2}t}{1-{\sin }^{2}t}\\ x^2-x^2{\sin }^{2}t-{\sin }^{2}t=0\\ {\sin }^{2}t=\frac{x^2}{1+x^2}\\ \sin t=\frac{x}{\sqrt{1+x^2}}\\ \\ {\displaystyle \int \frac{1}{{\left(x^2+1\right)}^{\frac{3}{2}}}dx}\\ ={\displaystyle \int \frac{1}{{\left({\tan }^{2}t+1\right)}^{\frac{3}{2}}}\frac{1}{{\cos }^{2}t}dt}\\ ={\displaystyle \int \frac{1}{{\left(\frac{1}{{\cos }^{2}t}\right)}^{\frac{3}{2}}}\frac{1}{{\cos }^{2}t}dt}\\ ={\displaystyle \int \cos tdt}\\ =\frac{x}{\sqrt{1+x^2}}\end{array}$
   3. 
      $\begin{array}{l}\alpha \ne -1\\ {\displaystyle \int \frac{{\left(\log x\right)}^{\alpha }}{x}dx}\\ =\frac{1}{\alpha +1}{\left(\log x\right)}^{\alpha +1}\\ \\ \alpha =-1\\ {\displaystyle \int \frac{1}{x\log x}dx}\\ =\log \left|\log x\right|\end{array}$
   4. 
      $\begin{array}{l}\sqrt{e^x-1}=t\\ e^x-1=t^2\\ e^x=t^2+1\\ x\log e=\log \left(t^2+1\right)\\ x=\log \left(t^2+1\right)\\ \frac{dx}{dt}=\frac{2t}{t^2+1}\\ \\ {\displaystyle \int \frac{2t^2}{t^2+1}dt}\\ ={\displaystyle \int \frac{2t^2+2-2}{t^2+1}dt}\\ ={\displaystyle \int \left(2-\frac{2}{t^2+1}\right)dt}\\ =2t-2{\displaystyle \int \frac{1}{t^2+1}dt}\\ =2t-2\arctan t\\ =2\left(\sqrt{e^x-1}-2\arctan \sqrt{e^x-1}\right)\end{array}$
   5. 
      $\begin{array}{l}\log x=t\\ e^t=x\\ \frac{dx}{dt}=e^t\\ \\ {\displaystyle \int e^t\sin tdt}\\ =e^t\sin t-{\displaystyle \int e^t\cos tdt}\\ =e^t\sin t-\left(e^t\cos t+{\displaystyle \int e^t\sin tdt}\right)\\ 2{\displaystyle \int e^t\sin tdt}=e^t\sin t-e^t\cos t\\ {\displaystyle \int e^t\sin tdt}\\ =\frac{e^t}{2}\left(\sin t-\cos t\right)\\ =\frac{e^{\log x}}{2}\left(\sin \left(\log x\right)-\cos \left(\log x\right)\right)\\ =\frac{x}{2}\left(\sin \left(\log x\right)-\cos \left(\log x\right)\right)\end{array}$
   6. 
      $\begin{array}{l}{\displaystyle \int \frac{\cos x}{a\cos x+b\sin x}}\\ ={\displaystyle \int \frac{\frac{a}{a^2+b^2}\left(a\cos x+b\sin x\right)+\frac{b}{a^2+b^2}\left(b\cos x-a\sin x\right)}{a\cos x+b\sin x}dx}\\ =={\displaystyle \int \left(\frac{a}{a^2+b^2}+\frac{\frac{b}{a^2+b^2}\left(b\cos x-a\sin x\right)}{a\cos x+b\sin x}\right)}dx\\ =\frac{a}{a^2+b^2}x+\frac{b}{a^2+b^2}\log \left|a\cos x+b\sin x\right|\end{array}$

コード(Emacs)

Python 3

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

from sympy import pprint, symbols, Integral, sin, cos, tan, exp, Rational, log, sqrt

print('8.')
x, α = symbols('x α')
a, b = symbols('a b', nonzero=True)

fs = [exp(-x) * cos(x) ** 2,
      1 / (x ** 2 + 1) ** Rational(3, 2),
      log(x) ** α / x,
      sqrt(exp(x) - 1),
      sin(log(x)),
      1 / (a + b * tan(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())


print('(3) α = -1')
f = log(x) ** (-1) / x
I = Integral(f, x)
pprint(I)
I = I.doit()
pprint(I)
print('factor:')
pprint(I.factor())
print('expand:')
pprint(I.expand())

print('(6)')
f = cos(x) / (a * cos(x) + b * sin(x))
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)

$ ./sample8.py
8.
(1)
⌠               
⎮  -x    2      
⎮ ℯ  ⋅cos (x) dx
⌡               
     -x    2         -x                    -x    2   
  2⋅ℯ  ⋅sin (x)   2⋅ℯ  ⋅sin(x)⋅cos(x)   3⋅ℯ  ⋅cos (x)
- ───────────── + ─────────────────── - ─────────────
        5                  5                  5      
factor:
 ⎛     2                             2   ⎞  -x 
-⎝2⋅sin (x) - 2⋅sin(x)⋅cos(x) + 3⋅cos (x)⎠⋅ℯ   
───────────────────────────────────────────────
                       5                       
expand:
     -x    2         -x                    -x    2   
  2⋅ℯ  ⋅sin (x)   2⋅ℯ  ⋅sin(x)⋅cos(x)   3⋅ℯ  ⋅cos (x)
- ───────────── + ─────────────────── - ─────────────
        5                  5                  5      
(2)
⌠               
⎮      1        
⎮ ─────────── dx
⎮         3/2   
⎮ ⎛ 2    ⎞      
⎮ ⎝x  + 1⎠      
⌡               
      1      
─────────────
     ________
    ╱     1  
   ╱  1 + ── 
  ╱        2 
╲╱        x  
factor:
      1      
─────────────
     ________
    ╱     1  
   ╱  1 + ── 
  ╱        2 
╲╱        x  
expand:
      1      
─────────────
     ________
    ╱     1  
   ╱  1 + ── 
  ╱        2 
╲╱        x  
(3)
⌠           
⎮    α      
⎮ log (x)   
⎮ ─────── dx
⎮    x      
⌡           
   α + 1   
log     (x)
───────────
   α + 1   
factor:
   α + 1   
log     (x)
───────────
   α + 1   
expand:
          α   
log(x)⋅log (x)
──────────────
    α + 1     
(4)
⌠               
⎮    ________   
⎮   ╱  x        
⎮ ╲╱  ℯ  - 1  dx
⌡               
⌠               
⎮    ________   
⎮   ╱  x        
⎮ ╲╱  ℯ  - 1  dx
⌡               
factor:
⌠               
⎮    ________   
⎮   ╱  x        
⎮ ╲╱  ℯ  - 1  dx
⌡               
expand:
⌠               
⎮    ________   
⎮   ╱  x        
⎮ ╲╱  ℯ  - 1  dx
⌡               
(5)
⌠               
⎮ sin(log(x)) dx
⌡               
x⋅sin(log(x))   x⋅cos(log(x))
───────────── - ─────────────
      2               2      
factor:
-x⋅(-sin(log(x)) + cos(log(x))) 
────────────────────────────────
               2                
expand:
x⋅sin(log(x))   x⋅cos(log(x))
───────────── - ─────────────
      2               2      
(6)
⌠                
⎮      1         
⎮ ──────────── dx
⎮ a + b⋅tan(x)   
⌡                
                         ⎛a         ⎞                    
                    b⋅log⎜─ + tan(x)⎟                    
  a⋅x      ⅈ⋅b⋅x         ⎝b         ⎠   b⋅log(tan(x) - ⅈ)
─────── + ─────── + ───────────────── - ─────────────────
 2    2    2    2         2    2              2    2     
a  + b    a  + b         a  + b              a  + b      
factor:
                   ⎛a         ⎞                    
a⋅x + ⅈ⋅b⋅x + b⋅log⎜─ + tan(x)⎟ - b⋅log(tan(x) - ⅈ)
                   ⎝b         ⎠                    
───────────────────────────────────────────────────
                       2    2                      
                      a  + b                       
expand:
                         ⎛a         ⎞                    
                    b⋅log⎜─ + tan(x)⎟                    
  a⋅x      ⅈ⋅b⋅x         ⎝b         ⎠   b⋅log(tan(x) - ⅈ)
─────── + ─────── + ───────────────── - ─────────────────
 2    2    2    2         2    2              2    2     
a  + b    a  + b         a  + b              a  + b      
(3) α = -1
⌠            
⎮    1       
⎮ ──────── dx
⎮ x⋅log(x)   
⌡            
log(log(x))
factor:
log(log(x))
expand:
log(log(x))
(6)
⌠                       
⎮        cos(x)         
⎮ ─────────────────── dx
⎮ a⋅cos(x) + b⋅sin(x)   
⌡                       
               ⎛         b⋅sin(x)⎞
          b⋅log⎜cos(x) + ────────⎟
  a⋅x          ⎝            a    ⎠
─────── + ────────────────────────
 2    2            2    2         
a  + b            a  + b          
factor:
           ⎛         b⋅sin(x)⎞
a⋅x + b⋅log⎜cos(x) + ────────⎟
           ⎝            a    ⎠
──────────────────────────────
            2    2            
           a  + b             
expand:
               ⎛         b⋅sin(x)⎞
          b⋅log⎜cos(x) + ────────⎟
  a⋅x          ⎝            a    ⎠
─────── + ────────────────────────
 2    2            2    2         
a  + b            a  + b          
$

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="sample8.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) => Math.exp(-x) * Math.cos(x) ** 2,
    f2 = (x) => 1 / Math.sqrt((x ** 2 + 1) ** 3),
    f3 = (x) => Math.log(x) ** 2 / x,
    f4 = (x) => Math.sqrt(Math.exp(x) - 1),
    f5 = (x) => Math.sin(Math.log(x)),
    f6 = (x) => 1 / (1 + 2 * Math.tan(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'],
               [f4, 'brown'],
               [f5, 'purple'],
               [f6, '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 =