数学 - Python - JavaScript - 細分による加法 - 積分法 – 不定積分の計算 - 置換積分法(三角関数(正弦、余弦、正接)、指数関数、対数関数、合成関数の微分法)

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)、置換積分法、問12.を取り組んでみる。

1. $\begin{array}{l} 1 + x^{2} = t \\ 2 x = \frac{d t}{d x} \\ \frac{1}{2} \int 2 x {(1 + x^{2})}^{3} d x \\ = \frac{1}{2} \int t^{3} \frac{d t}{d x} d x \\ = \frac{1}{2} \int t^{3} d t \\ = \frac{1}{2} \frac{1}{4} t^{4} \\ = \frac{1}{8} {(1 + x^{2})}^{4} \end{array}$
2. $\begin{array}{l} 1 + x^{3} = t \\ 3 x^{2} = \frac{d t}{d x} \\ \frac{1}{3} \int 3 x^{2} {(1 + x^{3})}^{4} d x \\ = \frac{1}{3} \int t^{4} d t \\ = \frac{1}{3} \frac{1}{5} t^{5} \\ = \frac{1}{15} {(1 + x^{3})}^{5} \end{array}$
3. $\begin{array}{l} \sin x = t \\ \cos x = \frac{d t}{d x} \\ \int t d t \\ = \frac{1}{2} t^{2} \\ = \frac{1}{2} \sin^{2} x \end{array}$
4. $\begin{array}{l} \sin x = t \\ \cos x = \frac{d t}{d x} \\ \int t^{2} d t \\ = \frac{1}{3} t^{3} \\ = \frac{1}{3} \sin^{3} x \end{array}$
5. $\begin{array}{l} \sin x = t \\ \cos x = \frac{d t}{d x} \\ \int t^{5} d t \\ = \frac{1}{6} t^{6} \\ = \frac{1}{6} \sin^{6} x \end{array}$
6. $\begin{array}{l} \cos x = t \\ - \sin x = \frac{d t}{d x} \\ - \int - \sin x \cos^{3} x d x \\ = - \int t^{3} d t \\ = - \frac{1}{4} t^{4} \\ = - \frac{1}{4} \cos^{4} x \end{array}$
7. $\begin{array}{l} x^{2} = t \\ 2 x = \frac{d t}{d x} \\ \frac{1}{2} \int 2 x e^{x^{2}} d x \\ = \frac{1}{2} \int e^{t} d t \\ = \frac{1}{2} e^{t} \\ = \frac{1}{2} e^{x^{2}} \end{array}$
8. $\begin{array}{l} \log x = t \\ \frac{1}{x} = \frac{d t}{d x} \\ \int t d t \\ = \frac{1}{2} t^{2} \\ = \frac{{(\log x)}^{2}}{2} \end{array}$
9. $\begin{array}{l} x^{2} + x + 1 = t \\ 2 x + 1 = \frac{d t}{d x} \\ \int \frac{1}{t} d t \\ = \log t \\ = \log (x^{2} + x + 1) \end{array}$
10. $\begin{array}{l} t = 1 - x^{4} \\ \frac{d t}{d x} = - 4 x^{3} \\ - \frac{1}{4} \int \frac{- 4 x^{3}}{1 - x^{4}} d x \\ = - \frac{1}{4} \int \frac{1}{t} d t \\ = - \frac{1}{4} \log | t | \\ = - \frac{1}{4} \log | 1 - x^{4} | \end{array}$
11. $\begin{array}{l} t = \cos x \\ \frac{d t}{d x} = - \sin x \\ - \int \frac{- \sin x}{\cos x} d x \\ = - \int \frac{1}{t} d t \\ = - \log | t | \\ = - \log | \cos x | \end{array}$
12. $\begin{array}{l} 1 + e^{x} = t \\ e^{x} = \frac{d t}{d x} \\ \int \frac{1}{t} d t \\ = \log t \\ = \log (1 + e^{x}) \end{array}$

コード(Emacs)

Python 3

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

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

print('12.')
x = symbols('x')
fs = [x * (1 + x ** 2) ** 3,
      x ** 2 * (1 + x ** 3) ** 4,
      sin(x) * cos(x),
      sin(x) ** 2 * cos(x),
      sin(x) ** 5 * cos(x),
      sin(x) * cos(x) ** 3,
      x * exp(x ** 2),
      log(x) / x,
      (2 * x + 1) / (x ** 2 + x + 1),
      x ** 3 / (1 - x ** 4),
      tan(x),
      exp(x) / (1 + exp(x))]


for i, f in enumerate(fs, 1):
    print(f'({i})')
    I = Integral(f, x)
    for o in [I, I.doit()]:
        pprint(o.factor())
        print()
    try:
        p = plot(f, show=False, legend=True)
        p.save(f'sample12_{i}.svg')
    except Exception as err:
        print(type(err), err)
    print()

入出力結果(Terminal, IPython)

$ ./sample12.py
12.
(1)
⌠               
⎮           3   
⎮   ⎛ 2    ⎞    
⎮ x⋅⎝x  + 1⎠  dx
⌡               

 2 ⎛ 2    ⎞ ⎛ 4      2    ⎞
x ⋅⎝x  + 2⎠⋅⎝x  + 2⋅x  + 2⎠
───────────────────────────
             8             


(2)
⌠                             
⎮                         4   
⎮  2        4 ⎛ 2        ⎞    
⎮ x ⋅(x + 1) ⋅⎝x  - x + 1⎠  dx
⌡                             

 3 ⎛ 12      9       6       3    ⎞
x ⋅⎝x   + 5⋅x  + 10⋅x  + 10⋅x  + 5⎠
───────────────────────────────────
                 15                


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

   2   
sin (x)
───────
   2   


(4)
⌠                  
⎮    2             
⎮ sin (x)⋅cos(x) dx
⌡                  

   3   
sin (x)
───────
   3   


(5)
⌠                  
⎮    5             
⎮ sin (x)⋅cos(x) dx
⌡                  

   6   
sin (x)
───────
   6   


(6)
⌠                  
⎮           3      
⎮ sin(x)⋅cos (x) dx
⌡                  

    4    
-cos (x) 
─────────
    4    


(7)
⌠           
⎮    ⎛ 2⎞   
⎮    ⎝x ⎠   
⎮ x⋅ℯ     dx
⌡           

 ⎛ 2⎞
 ⎝x ⎠
ℯ    
─────
  2  


(8)
⌠          
⎮ log(x)   
⎮ ────── dx
⎮   x      
⌡          

   2   
log (x)
───────
   2   


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

   ⎛ 2        ⎞
log⎝x  + x + 1⎠


(10)
 ⌠                            
 ⎮             3              
 ⎮            x               
-⎮ ──────────────────────── dx
 ⎮                 ⎛ 2    ⎞   
 ⎮ (x - 1)⋅(x + 1)⋅⎝x  + 1⎠   
 ⌡                            

    ⎛ 4    ⎞ 
-log⎝x  - 1⎠ 
─────────────
      4      


(11)
⌠          
⎮ tan(x) dx
⌡          

    ⎛   2       ⎞ 
-log⎝sin (x) - 1⎠ 
──────────────────
        2         


(12)
⌠          
⎮    x     
⎮   ℯ      
⎮ ────── dx
⎮  x       
⎮ ℯ  + 1   
⌡          

   ⎛ x    ⎞
log⎝ℯ  + 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">
<br>
<label for="a0">a = </label>
<input id="a0" type="number" value="2">
<label for="b0">b = </label>
<input id="b0" type="number" value="3">
<label for="n0">n = </label>
<input id="n0" type="number" value="4">


<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="sample12.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 f2 = (x) => x ** 2 * (1 + x ** 3) ** 4,
    f7 = (x) => x * Math.exp(x ** 2),
    f8 = (x) => Math.log(x) / x,
    f10 = (x) => Math.tan(x),
    f11 = (x) => Math.exp(x) / (1 + Math.exp(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 || a0 <= 0 || a0 === 0 ||
        n0 === -1) {
        return;
    }
    
    let points = [],
        lines = [],
        fns = [[f2, 'red'],
               [f7, 'green'],
               [f8, 'blue'],
               [f10, 'orange'],
               [f11, '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 =
a = b = n =

Kamimura's blog

2017年8月16日水曜日

数学 - Python - JavaScript - 細分による加法 - 積分法 – 不定積分の計算 - 置換積分法(三角関数(正弦、余弦、正接)、指数関数、対数関数、合成関数の微分法)

0 コメント:

コメントを投稿