数学 - Python - JavaScript - 細分による加法 - 積分法 – 不定積分の計算 - 三角関数、指数関数の積分(対数関数)

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)、三角関数、指数関数の積分、問9、10.を取り組んでみる。

1. $- 3 \cos x - 4 \sin x$
2. $\begin{array}{l} \int (\cos x \tan x + 2 \cos x) d x \\ = \int (\sin x + 2 \cos x) d x \\ = - \cos x + 2 \sin x \end{array}$
3. $\begin{array}{l} \int (\frac{1}{\cos^{2} x} - \cos x) d x \\ = \tan x - \sin x \end{array}$
4. $4 e^{x} + 5 x$
$\frac{a^{x}}{\log a}$

コード(Emacs)

Python 3

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

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

print('9.')
x = symbols('x')
fs = [3 * sin(x) - 4 * cos(x),
      (tan(x) + 2) * cos(x),
      (1 - cos(x) ** 3) / cos(x) ** 2,
      4 * exp(x) + 5]


for i, f in enumerate(fs, 1):
    print(f'({i})')
    I = Integral(f, x)
    for o in [I, I.doit()]:
        pprint(o)
        print()
    p = plot(f, show=False, legend=True)
    p.save(f'sample9_{i}.svg')
    print()

print('10.')
a = symbols('a', positive=True)
f = a ** x
I = Integral(f, x)
for o in [I, I.doit()]:
    pprint(o)
    print()
p = plot(f.subs({a: 2}), show=False, legend=True)
p.save(f'sample10.svg')

入出力結果(Terminal, IPython)

$ ./sample9.py
9.
(1)
⌠                         
⎮ (3⋅sin(x) - 4⋅cos(x)) dx
⌡                         

-4⋅sin(x) - 3⋅cos(x)


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

2⋅sin(x) - cos(x)


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

      3⎛x⎞ 
-4⋅tan ⎜─⎟ 
       ⎝2⎠ 
───────────
   4⎛x⎞    
tan ⎜─⎟ - 1
    ⎝2⎠    


(4)
⌠              
⎮ ⎛   x    ⎞   
⎮ ⎝4⋅ℯ  + 5⎠ dx
⌡              

         x
5⋅x + 4⋅ℯ 


10.
⌠      
⎮  x   
⎮ a  dx
⌡      

⎧  x     for log(a) = 0
⎪                      
⎪   x                  
⎨  a                   
⎪──────    otherwise   
⎪log(a)                
⎩                      

$

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">
<br>
<label for="a0">a = </label>
<input id="a0" type="number" min="0" step="0.1" value="0.5">


<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="sample9.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'),
    input_a0 = document.querySelector('#a0'),
    inputs = [input_r, input_dx, input_x1, input_x2, input_y1, input_y2,
              input_a0],
    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 f9 = (x) => (1 - Math.cos(x) ** 3) / Math.cos(x) ** 2;
    
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),
        a0 = parseFloat(input_a0.value);

    if (r === 0 || dx === 0 || x1 > x2 || y1 > y2 || a0 <= 0 || a0 === 1) {
        return;
    }
    
    let points = [],
        lines = [],
        f10 = (x) => a0 ** x,
        fns = [[f9, 'green'],
               [f10, 'orange']],
        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 =

Kamimura's blog

2017年8月14日月曜日

数学 - Python - JavaScript - 細分による加法 - 積分法 – 不定積分の計算 - 三角関数、指数関数の積分(対数関数)

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