数学 - Python - JavaScript - 細分による加法 - 積分法 – 不定積分の計算 - 分数関数の積分(三角関数(逆正接)、部分分数式に分解、置換積分法、対数関数、指数関数)

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

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)、分数関数の積分、問18.を取り組んでみる。

1. $\begin{array}{l} \int (2 x + 2 + \frac{2}{x - 1}) d x \\ = x^{2} + 2 x + 2 \log | x - 1 | \end{array}$
2. $\begin{array}{l} \int \frac{(x + 2) (x^{2} - 2 x + 3) - 4}{x + 2} d x \\ = \int (x^{2} - 2 x + 3 - \frac{4}{x + 2}) d x \\ = \frac{1}{3} x^{3} - x^{2} + 3 x - 4 \log | x + 2 | \end{array}$
3. $\begin{array}{l} \frac{a}{x - 1} + \frac{b}{x - 2} \\ = \frac{(a + b) x - 2 a - b}{(x - 1) (x - 2)} \\ a + b = 1 \\ - 2 a - b = - 3 \\ b = 1 - a \\ - 2 a - a + a = - 3 \\ a = \frac{3}{2} \\ b = - \frac{1}{2} \\ \int (\frac{3}{2 (x - 1)} - \frac{1}{2 (x - 2)}) d x \\ = \frac{3}{2} \log | x - 1 | - \frac{1}{2} \log | x - 2 | \end{array}$
4. $\begin{array}{l} \int \frac{1}{x^{2} + {(- 2)}^{2}} d x \\ = - \frac{1}{2} \arctan (- \frac{x}{2}) \\ = \frac{1}{2} \arctan \frac{x}{2} \end{array}$
5. $\begin{array}{l} \int \frac{(x^{2} + x - 2) (x + 2) + 4}{x^{2} + x - 2} d x \\ = \int (x + 2 + \frac{4}{x^{2} + x - 2}) d x \\ = \frac{1}{2} x^{2} + 2 x + \int \frac{4}{(x + 2) (x - 1)} d x \\ \frac{a}{x + 2} + \frac{b}{x - 1} \\ = \frac{(a + b) x + (- a + 2 b)}{(x + 2) (x - 1)} \\ a + b = 0 \\ - a + 2 b = 4 \\ b = - a \\ - a - 2 a = 4 \\ a = - \frac{4}{3} \\ b = \frac{4}{3} \\ \frac{1}{2} x^{2} + 2 x + \int \frac{4}{(x + 2) (x - 1)} d x \\ = \frac{1}{2} x^{2} + 2 x + \int (- \frac{4}{3} \frac{1}{x + 2} + \frac{4}{3} \frac{1}{x - 1}) d x \\ = \frac{1}{2} x^{2} + 2 x - \frac{4}{3} \log | x + 2 | + \frac{4}{3} \log | x - 1 | \end{array}$
6. $\begin{array}{l} \frac{a}{x} + \frac{b}{x - 1} + \frac{c}{{(x - 1)}^{2}} \\ = \frac{a x^{2} - 2 a x + a + b x^{2} - b x + c x}{x {(x - 1)}^{2}} \\ = \frac{(a + b) x^{2} + (- 2 a - b + c) x + a}{x {(x - 1)}^{2}} \\ a + b = 0 \\ - a - b + c = 2 \\ a = 1 \\ b = - 1 \\ - 1 + 1 + c = 2 \\ c = 2 \\ \int (\frac{1}{x} - \frac{1}{x - 1} + \frac{2}{{(x - 1)}^{2}}) d x \\ = \log | x | - \log | x - 1 | - \frac{2}{x - 1} \end{array}$
7. $\begin{array}{l} \int \frac{1}{{(x + \frac{1}{2})}^{2} + {(\frac{\sqrt{3}}{2})}^{2}} d x \\ t = x + \frac{1}{2} \\ \frac{d t}{d x} = 1 \\ \int \frac{1}{{(x + \frac{1}{2})}^{2} + {(\frac{\sqrt{3}}{2})}^{2}} d x \\ = \int \frac{1}{t^{2} + {(\frac{\sqrt{3}}{2})}^{2}} d t \\ = \frac{2}{\sqrt{3}} \arctan \frac{\sqrt{3}}{2} t \\ = \frac{2}{\sqrt{3}} \arctan (\frac{\sqrt{3}}{2} (x + \frac{1}{2})) \end{array}$
8. $\begin{array}{l} \int \frac{x}{{(x + \frac{1}{2})}^{2} + {(\frac{\sqrt{3}}{2})}^{2}} d x \\ t = x + \frac{1}{2} \\ \frac{d t}{d x} = 1 \\ x = t - \frac{1}{2} \\ \int \frac{x}{{(x + \frac{1}{2})}^{2} + {(\frac{\sqrt{3}}{2})}^{2}} d x \\ = \int \frac{t - \frac{1}{2}}{t^{2} + {(\frac{\sqrt{3}}{2})}^{2}} d t \\ = \int (\frac{t}{t^{2} + {(\frac{\sqrt{3}}{2})}^{2}} - \frac{1}{2} \cdot \frac{1}{t^{2} + {(\frac{\sqrt{3}}{2})}^{2}}) d t \\ = \frac{1}{2} \log (t^{2} + {(\frac{\sqrt{3}}{2})}^{2}) - \frac{1}{2} \cdot \frac{2}{\sqrt{3}} \arctan \frac{2}{\sqrt{3}} t \\ = \frac{1}{2} \log (x^{2} + x + 1) - \frac{1}{\sqrt{3}} \arctan (\frac{2}{\sqrt{3}} (x + \frac{1}{2})) \end{array}$
9. $\begin{array}{l} \frac{a}{x + 1} + \frac{b}{{(x + 1)}^{2}} + \frac{c}{{(x + 1)}^{3}} \\ = \frac{a x^{2} + 2 a x + a + b x + b + c}{{(x + 1)}^{3}} \\ = \frac{a x^{2} + (2 a + b) x + a + b + c}{{(x + 1)}^{3}} \\ a = 0 \\ 2 a + b = 1 \\ a + b + c = 0 \\ b = 1 \\ c = - 1 \\ \int (\frac{1}{{(x + 1)}^{2}} - \frac{1}{{(x + 1)}^{3}}) d x \\ = - \frac{1}{x + 1} - \frac{1}{2} {(x + 1)}^{- 2} \end{array}$
10. $\begin{array}{l} \frac{a}{x} + \frac{b x + c}{x^{2} + 1} + \frac{d x + e}{{(x^{2} + 1)}^{2}} \\ = \frac{a x^{4} + 2 a x^{2} + a + b x^{4} + b x^{2} + c x^{3} + c x + d x^{2} + e x}{x (x^{2} + 1)} \\ = \frac{(a + b) x^{4} + c x^{3} + (2 a + b + d) x^{2} + (c + e) x + a}{x (x^{2} + 1)} \\ a + b = 0 \\ c = 0 \\ 2 a + b + d = 0 \\ c + e = 0 \\ a = 1 \\ b = - 1 \\ 2 - 1 + d = 0 \\ d = - 1 \\ e = 0 \\ \int (\frac{1}{x} + \frac{- x}{x^{2} + 1} + \frac{- x}{{(x^{2} + 1)}^{2}}) d x \\ = \log | x | - \frac{1}{2} \log (x^{2} + 1) + \frac{1}{2} \cdot \frac{1}{x^{2} + 1} \end{array}$

コード(Emacs)

Python 3

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

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

print('18.')
x = symbols('x')
fs = [2 * x ** 2 / (x - 1),
      (x ** 3 - x + 2) / (x + 2),
      (x - 3) / ((x - 1) * (x - 2)),
      1 / (x ** 2 - 4),
      x ** 3 / (x ** 2 + x - 2),
      (2 * x + 1) / (x * (x - 1) ** 2),
      1 / (x ** 2 + x + 1),
      x / (x ** 2 + x + 1),
      x / (x + 1) ** 3,
      1 / (x * (x ** 2 + 1) ** 2)]

for i, f in enumerate(fs, 1):
    print(f'({i})')
    I = Integral(f, x)
    pprint(I)
    I = I.doit()
    pprint(I)
    for j, g in enumerate([f, I]):
        p = plot(g, show=False, legend=True)
        p.save(f'sample18_{i}_{j}.svg')
    print()

入出力結果(Terminal, IPython)

$ ./sample18.py
18.
(1)
⌠         
⎮     2   
⎮  2⋅x    
⎮ ───── dx
⎮ x - 1   
⌡         
 2                     
x  + 2⋅x + 2⋅log(x - 1)

(2)
⌠              
⎮  3           
⎮ x  - x + 2   
⎮ ────────── dx
⎮   x + 2      
⌡              
 3                          
x     2                     
── - x  + 3⋅x - 4⋅log(x + 2)
3                           

(3)
⌠                   
⎮      x - 3        
⎮ ─────────────── dx
⎮ (x - 2)⋅(x - 1)   
⌡                   
-log(x - 2) + 2⋅log(x - 1)

(4)
⌠          
⎮   1      
⎮ ────── dx
⎮  2       
⎮ x  - 4   
⌡          
log(x - 2)   log(x + 2)
────────── - ──────────
    4            4     

(5)
⌠              
⎮      3       
⎮     x        
⎮ ────────── dx
⎮  2           
⎮ x  + x - 2   
⌡              
 2                                
x        log(x - 1)   8⋅log(x + 2)
── - x + ────────── + ────────────
2            3             3      

(6)
⌠              
⎮  2⋅x + 1     
⎮ ────────── dx
⎮          2   
⎮ x⋅(x - 1)    
⌡              
                        3  
log(x) - log(x - 1) - ─────
                      x - 1

(7)
⌠              
⎮     1        
⎮ ────────── dx
⎮  2           
⎮ x  + x + 1   
⌡              
         ⎛2⋅√3⋅x   √3⎞
2⋅√3⋅atan⎜────── + ──⎟
         ⎝  3      3 ⎠
──────────────────────
          3           

(8)
⌠              
⎮     x        
⎮ ────────── dx
⎮  2           
⎮ x  + x + 1   
⌡              
                         ⎛2⋅√3⋅x   √3⎞
   ⎛ 2        ⎞   √3⋅atan⎜────── + ──⎟
log⎝x  + x + 1⎠          ⎝  3      3 ⎠
─────────────── - ────────────────────
       2                   3          

(9)
⌠            
⎮    x       
⎮ ──────── dx
⎮        3   
⎮ (x + 1)    
⌡            
 -(2⋅x + 1)   
──────────────
   2          
2⋅x  + 4⋅x + 2

(10)
⌠               
⎮      1        
⎮ ─────────── dx
⎮           2   
⎮   ⎛ 2    ⎞    
⎮ x⋅⎝x  + 1⎠    
⌡               
            ⎛ 2    ⎞           
         log⎝x  + 1⎠      1    
log(x) - ─────────── + ────────
              2           2    
                       2⋅x  + 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="sample18.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) => 2 * x ** 2 / (x - 1),
    f2 = (x) => (x ** 3 - x + 2) / (x + 2),
    f3 = (x) => (x - 3) / ((x - 1) * (x - 2)),
    f4 = (x) => 1 / (x ** 2 - 4),
    f5 = (x) => x ** 3 / (x ** 2 + x - 1);    

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, 'orange'],
               [f5, 'brown']],
        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 =

Kamimura's blog

2017年8月22日火曜日

数学 - Python - JavaScript - 細分による加法 - 積分法 – 不定積分の計算 - 分数関数の積分(三角関数(逆正接)、部分分数式に分解、置換積分法、対数関数、指数関数)

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