数学 - Python - JavaScript - 解析学 - 積分 - 積分の計算 - 三角関数に関する積分(置換積分法、部分分数分解、対数関数、微分、不定積分)

学習環境

解析入門原書第3版 (S.ラング(著)、松坂和夫(翻訳)、片山孝次(翻訳)、岩波書店)の第3部(積分)、第11章(積分の計算)、補充問題(三角関数に関する積分)24.を取り組んでみる。

$\begin{array}{l} t = \sqrt{a^{2} + x^{2}} \\ \frac{dt}{dx} = \frac{x}{\sqrt{a^{2} + x^{2}}} \\ \int \frac{\sqrt{a^{2} + x^{2}}}{x} dx \\ = \int \frac{t}{x} \cdot \frac{\sqrt{a^{2} + x^{2}}}{x} dt \\ = \int \frac{t^{2}}{x^{2}} dt \\ = \int \frac{t^{2}}{t^{2} - a^{2}} dt \\ = \int 1 dt + a^{2} \int \frac{1}{t^{2} - a^{2}} dt \\ \frac{A}{t + a} + \frac{B}{t - a} = \frac{(A + B) t + a (B - A)}{t^{2} - a^{2}} \\ A + B = 0 \\ - A + B = \frac{1}{a} \\ B = \frac{1}{2 a} \\ A = - \frac{1}{2 a} \\ \int \frac{1}{t^{2} - a^{2}} dt \\ = \frac{1}{2 a} (- \log (t + a) + \log (t - a)) \\ = \frac{1}{2 a} \log \frac{t - a}{t + a} \end{array}$

よって、求める不定積分は、

$\begin{array}{l} t + \frac{a}{2} \log \frac{t - a}{t + a} \\ = \sqrt{a^{2} + x^{2}} + \frac{a}{2} \log \frac{\sqrt{a^{2} + x^{2}} - a}{\sqrt{a^{2} + x^{2}} + a} \\ = \sqrt{a^{2} + x^{2}} + \frac{a}{2} \log \frac{{(\sqrt{a^{2} + x^{2}} - a)}^{2}}{x^{2}} \\ = \sqrt{a^{2} + x^{2}} + a \log \frac{\sqrt{a^{2} + x^{2}} - a}{x} \end{array}$

コード(Emacs)

Python 3

#!/usr/bin/env python3
from sympy import pprint, symbols, Integral, plot, sqrt, log, Derivative

print('24.')

a, x = symbols('a, x')
f = sqrt(a ** 2 + x ** 2) / x
I = Integral(f, x)
for t in [I, I.doit().simplify()]:
    pprint(t)
    print()

a0 = 2
p = plot(f.subs({a: a0}), I.doit().simplify().subs(
    {a: a0}), legend=True, show=False)
colors = ['red', 'green']
for i, c in enumerate(colors):
    p[i].line_color = c
p.save('sample24.svg')

g = sqrt(a ** 2 + x ** 2) + a * log((sqrt(a ** 2 + x ** 2) - a) / x)
d = Derivative(g, x, 1)
for t in [d, d.doit().factor()]:
    pprint(t)
    print()

入出力結果(Terminal, Jupyter(IPython))

$ ./sample24.py
24.
⌠                
⎮    _________   
⎮   ╱  2    2    
⎮ ╲╱  a  + x     
⎮ ──────────── dx
⎮      x         
⌡                

                       ________
                      ╱  2     
         ⎛a⎞         ╱  a      
- a⋅asinh⎜─⎟ + x⋅   ╱   ── + 1 
         ⎝x⎠       ╱     2     
                 ╲╱     x      

  ⎛     ⎛        _________⎞               ⎞
  ⎜     ⎜       ╱  2    2 ⎟      _________⎟
∂ ⎜     ⎜-a + ╲╱  a  + x  ⎟     ╱  2    2 ⎟
──⎜a⋅log⎜─────────────────⎟ + ╲╱  a  + x  ⎟
∂x⎝     ⎝        x        ⎠               ⎠

   _________
  ╱  2    2 
╲╱  a  + x  
────────────
     x      

$

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="0.001" 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="sample24.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_n0 = document.querySelector('#n0'),
    inputs = [input_r, input_dx, input_x1, input_x2, input_y1, input_y2],
    p = (x) => pre0.textContent += x + '\n';

let f = (x) => Math.sqrt(2 ** 2 + x ** 2) / x,
    g = (x) => Math.sqrt(2 ** 2 + x ** 2) + 2 * Math.log((Math.sqrt(2 ** 2 + x ** 2) - 2) / x),
    fns = [[f, 'red'],
           [g, 'green']];

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
        .forEach((o) => {
            let [f, color] = o;
            for (let x = x1; x <= x2; x += dx) {
                let y = f(x);

                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('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].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 =

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数学 - Python - JavaScript - 解析学 - 積分 - 積分の計算 - 三角関数に関する積分(置換積分法、部分分数分解、対数関数、微分、不定積分)

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