数学 - Python - JavaScript - 細分による加法 - 積分法 – 不定積分の計算 - 部分積分法(三角関数(正弦、余弦)、指数関数、対数関数、積の微分法)

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)、部分積分法、問14.を取り組んでみる。

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

コード(Emacs)

Python 3

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

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

print('14.')
x = symbols('x')
fs = [x * sin(x),
      x * cos(x),
      x * exp(x),
      x * exp(-x),
      x * exp(-2 * x),
      x * log(x),
      x ** 2 * log(x),
      log(x) ** 2,
      x ** 2 * sin(x),
      x ** 2 * exp(x),
      exp(-x) * sin(x),
      exp(2 * x) * cos(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'sample14_{i}.svg')
    except Exception as err:
        print(type(err), err)
    print()

入出力結果(Terminal, IPython)

$ ./sample14.py
14.
(1)
⌠            
⎮ x⋅sin(x) dx
⌡            

-x⋅cos(x) + sin(x)


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

x⋅sin(x) + cos(x)


(3)
⌠        
⎮    x   
⎮ x⋅ℯ  dx
⌡        

         x
(x - 1)⋅ℯ 


(4)
⌠         
⎮    -x   
⎮ x⋅ℯ   dx
⌡         

          -x
-(x + 1)⋅ℯ  


(5)
⌠           
⎮    -2⋅x   
⎮ x⋅ℯ     dx
⌡           

            -2⋅x 
-(2⋅x + 1)⋅ℯ     
─────────────────
        4        


(6)
⌠            
⎮ x⋅log(x) dx
⌡            

 2               
x ⋅(2⋅log(x) - 1)
─────────────────
        4        


(7)
⌠             
⎮  2          
⎮ x ⋅log(x) dx
⌡             

 3               
x ⋅(3⋅log(x) - 1)
─────────────────
        9        


(8)
⌠           
⎮    2      
⎮ log (x) dx
⌡           

  ⎛   2                  ⎞
x⋅⎝log (x) - 2⋅log(x) + 2⎠


(9)
⌠             
⎮  2          
⎮ x ⋅sin(x) dx
⌡             

   2                               
- x ⋅cos(x) + 2⋅x⋅sin(x) + 2⋅cos(x)


(10)
⌠         
⎮  2  x   
⎮ x ⋅ℯ  dx
⌡         

⎛ 2          ⎞  x
⎝x  - 2⋅x + 2⎠⋅ℯ 


(11)
⌠              
⎮  -x          
⎮ ℯ  ⋅sin(x) dx
⌡              

                    -x 
-(sin(x) + cos(x))⋅ℯ   
───────────────────────
           2           


(12)
⌠               
⎮  2⋅x          
⎮ ℯ   ⋅cos(x) dx
⌡               

                     2⋅x
(sin(x) + 2⋅cos(x))⋅ℯ   
────────────────────────
           5            


$

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="sample14.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 f3 = (x) => x * Math.exp(x),
    f4 = (x) => x * Math.exp(-x),
    f5 = (x) => x * Math.exp(-2 * x),
    f10 = (x) => x ** 2 * Math.exp(x),
    f11 = (x) => Math.exp(-x) * Math.sin(x),
    f12 = (x) => Math.exp(2 * x) * Math.cos(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 = [[f3, 'red'],
               [f4, 'green'],
               [f5, 'blue'],
               [f10, 'orange'],
               [f11, 'brown'],
               [f12, 'purple']],
        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18日金曜日

数学 - Python - JavaScript - 細分による加法 - 積分法 – 不定積分の計算 - 部分積分法(三角関数(正弦、余弦)、指数関数、対数関数、積の微分法)

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