## 2018年10月4日木曜日

### 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)

1. $\begin{array}{}t=\sqrt{{a}^{2}+{x}^{2}}\\ \frac{\mathrm{dt}}{\mathrm{dx}}=\frac{x}{\sqrt{{a}^{2}+{x}^{2}}}\\ \int \frac{\sqrt{{a}^{2}+{x}^{2}}}{x}\mathrm{dx}\\ =\int \frac{t}{x}·\frac{\sqrt{{a}^{2}+{x}^{2}}}{x}\mathrm{dt}\\ =\int \frac{{t}^{2}}{{x}^{2}}\mathrm{dt}\\ =\int \frac{{t}^{2}}{{t}^{2}-{a}^{2}}\mathrm{dt}\\ =\int 1\mathrm{dt}+{a}^{2}\int \frac{1}{{t}^{2}-{a}^{2}}\mathrm{dt}\\ \frac{A}{t+a}+\frac{B}{t-a}=\frac{\left(A+B\right)t+a\left(B-A\right)}{{t}^{2}-{a}^{2}}\\ A+B=0\\ -A+B=\frac{1}{a}\\ B=\frac{1}{2a}\\ A=-\frac{1}{2a}\\ \int \frac{1}{{t}^{2}-{a}^{2}}\mathrm{dt}\\ =\frac{1}{2a}\left(-\mathrm{log}\left(t+a\right)+\mathrm{log}\left(t-a\right)\right)\\ =\frac{1}{2a}\mathrm{log}\frac{t-a}{t+a}\end{array}$

よって、求める不定積分は、

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

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()


$./sample24.py 24. ⌠ ⎮ _________ ⎮ ╱ 2 2 ⎮ ╲╱ a + x ⎮ ──────────── dx ⎮ x ⌡ ________ ╱ 2 ⎛a⎞ ╱ a - a⋅asinh⎜─⎟ + x⋅ ╱ ── + 1 ⎝x⎠ ╱ 2 ╲╱ x ⎛ ⎛ _________⎞ ⎞ ⎜ ⎜ ╱ 2 2 ⎟ _________⎟ ∂ ⎜ ⎜-a + ╲╱ a + x ⎟ ╱ 2 2 ⎟ ──⎜a⋅log⎜─────────────────⎟ + ╲╱ a + x ⎟ ∂x⎝ ⎝ x ⎠ ⎠ _________ ╱ 2 2 ╲╱ a + x ──────────── x$


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,
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])
let yscale = d3.scaleLinear()
.domain([y1, y2])

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();