## 2018年6月7日木曜日

### 数学 - Python - JavaScript - 解析学 - 積分 - 積分の計算 - 三角関数の積分(平方根、逆数、正弦、余弦、逆三角関数(逆正弦関数))

1. $\begin{array}{}x=3\mathrm{sin}t\\ \frac{\mathrm{dx}}{\mathrm{dt}}=3\mathrm{cos}t\\ \int \frac{1}{\sqrt{9-9{\mathrm{sin}}^{2}t}}·3\mathrm{cos}t\mathrm{dt}\\ =\int 1\mathrm{dt}\\ =t\\ =\mathrm{arcsin}\frac{x}{3}\end{array}$

2. $\begin{array}{}x=\sqrt{3}\mathrm{sin}t\\ \frac{\mathrm{dx}}{\mathrm{dt}}=\sqrt{3}\mathrm{cos}t\\ \int \frac{1}{\sqrt{3-3{\mathrm{sin}}^{2}t}}\sqrt{3}\mathrm{cos}t\mathrm{dt}\\ =\int A\mathrm{dt}\\ =t\\ =\mathrm{arcsin}\frac{x}{\sqrt{3}}\end{array}$

3. $\begin{array}{}x=\frac{1}{\sqrt{2}}\mathrm{sin}t\\ \frac{\mathrm{dx}}{\mathrm{dt}}=\frac{1}{\sqrt{2}}\mathrm{cos}t\\ \int \frac{1}{\sqrt{2-2{\mathrm{sin}}^{2}t}}·\frac{1}{\sqrt{2}}\mathrm{cos}t\mathrm{dt}\\ =\frac{1}{2}\int 1\mathrm{dt}\\ =\frac{1}{2}t\\ =\frac{1}{2}\mathrm{arcsin}\left(\sqrt{2}x\right)\end{array}$

4. $\begin{array}{}x=\frac{a}{b}\mathrm{sin}t\\ \frac{\mathrm{dx}}{\mathrm{dt}}=\frac{a}{b}\mathrm{cos}t\\ \int \frac{1}{a\sqrt{1-{\mathrm{sin}}^{2}t}}\frac{a}{b}\mathrm{cos}t\mathrm{dt}\\ =\frac{1}{b}\int 1\mathrm{dt}\\ =\frac{1}{b}t\\ =\frac{1}{b}\mathrm{arcsin}\left(\frac{b}{a}x\right)\end{array}$

コード(Emacs)

Python 3

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

x, a, b = symbols('x, a, b')
fs = [1 / sqrt(9 - x ** 2),
1 / sqrt(3 - x ** 2),
1 / sqrt(2 - 4 * x ** 2),
1 / sqrt(a ** 2 - b ** 2 * x ** 2)]

for i, f in enumerate(fs, 14):
print(f'{i}.')
I = Integral(f, x)
for t in [I, I.doit()]:
pprint(t)
print()

p = plot(*fs[:3], legend=True, show=False)
for i, color in enumerate(['red', 'green', 'blue']):
p[i].line_color = color
p.save('sample14.svg')


$./sample14.py 14. ⌠ ⎮ 1 ⎮ ───────────── dx ⎮ __________ ⎮ ╱ 2 ⎮ ╲╱ - x + 9 ⌡ ⎛x⎞ asin⎜─⎟ ⎝3⎠ 15. ⌠ ⎮ 1 ⎮ ───────────── dx ⎮ __________ ⎮ ╱ 2 ⎮ ╲╱ - x + 3 ⌡ ⎛√3⋅x⎞ asin⎜────⎟ ⎝ 3 ⎠ 16. ⌠ ⎮ 1 ⎮ ─────────────── dx ⎮ ____________ ⎮ ╱ 2 ⎮ ╲╱ - 4⋅x + 2 ⌡ ⎧-ⅈ⋅acosh(√2⋅x) │ 2│ ⎪─────────────── for 2⋅│x │ > 1 ⎪ 2 ⎨ ⎪ asin(√2⋅x) ⎪ ────────── otherwise ⎩ 2 17. ⌠ ⎮ 1 ⎮ ─────────────── dx ⎮ ____________ ⎮ ╱ 2 2 2 ⎮ ╲╱ a - b ⋅x ⌡ ⎧ ⎛b⋅x⎞ ⎪-ⅈ⋅acosh⎜───⎟ │ 2 2│ ⎪ ⎝ a ⎠ │b ⋅x │ ⎪────────────── for │─────│ > 1 ⎪ b │ 2 │ ⎪ │ a │ ⎨ ⎪ ⎛b⋅x⎞ ⎪ asin⎜───⎟ ⎪ ⎝ a ⎠ ⎪ ───────── otherwise ⎪ b ⎩$


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">
<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">

<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,
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'),
input_b0 = document.querySelector('#b0'),
inputs = [input_r, input_dx, input_x1, input_x2, input_y1, input_y2,
input_a0, input_b0],
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 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),
b0 = parseFloat(input_b0.value);

if (r === 0 || dx === 0 || x1 > x2 || y1 > y2) {
return;
}

let points = [],
lines = [[-Math.PI, y1, -Math.PI, y2, 'orange'],
[Math.PI, y1, Math.PI, y2, 'brown']],
fns = [[(x) => 1 / Math.sqrt(9 - x ** 2), 'red'],
[(x) => 1 / Math.sqrt(3 - x ** 2), 'green'],
[(x) => 1 / Math.sqrt(2 - 4 * x ** 2), 'blue'],
[(x) => 1 / Math.sqrt(a0 ** 2 - b0 ** 2 * x ** 2), 'purple']],
fns1 = [],
fns2 = [];

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

points.push([x, y, color]);
}
});

fns1
.forEach((o) => {
let [f, color] = o;

lines.push([x1, f(x1), x2, f(x2), color]);
});

fns2
.forEach((o) => {
let [f, color] = o;
for (let x = x1; x <= x2; x += dx0) {
let g = f(x);
lines.push([x1, g(x1), x2, g(x2), 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, fns1, fns2].forEach((fs) => p(fs.join('\n')));
};

inputs.forEach((input) => input.onchange = draw);
btn0.onclick = draw;
btn1.onclick = () => pre0.textContent = '';
draw();