## 2018年9月24日月曜日

### 数学 - Python - JavaScript - 解析学 - 積分 - 積分の計算 - 三角関数に関する積分(部分積分方、置換積分法、正弦、余弦、逆正弦、加法定理、2倍角、不定積分)

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

コード(Emacs)

Python 3

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

print('17.')

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

a0 = 2
p = plot(f.subs({a: a0}), I.doit().subs({a: a0}), legend=True, show=False)
colors = ['red', 'green']
for i, c in enumerate(colors):
p[i].line_color = c
p.save('sample17.svg')

g = -x * sqrt(a ** 2 - x ** 2) + a / 2 * x * \
cos(asin(x / a)) + a ** 2 / 2 * asin(x / a)

d = Derivative(g, x, 1)
for t in [d, d.doit().simplify()]:
pprint(t)
print()

pprint((d.doit() - x ** 2 / sqrt(a ** 2 - x ** 2)).simplify())


$./sample17.py 17. ⌠ ⎮ 2 ⎮ x ⎮ ──────────── dx ⎮ _________ ⎮ ╱ 2 2 ⎮ ╲╱ a - x ⌡ ⎧ ⎛ _____________⎞ ⎪ ⎜ ╱ ⎛ 2 2⎞ ⎟ ⎪ ⎜ ⎛x⎞ ╱ -⎝a - x ⎠ ⎟ ⎪-ⅈ⋅a⋅⎜a⋅acosh⎜─⎟ + x⋅ ╱ ─────────── ⎟ ⎪ ⎜ ⎝a⎠ ╱ 2 ⎟ │ 2│ ⎪ ⎝ ╲╱ a ⎠ │x │ ⎪────────────────────────────────────────── for │──│ > 1 ⎪ 2 │ 2│ ⎪ │a │ ⎨ ⎪ ⎛ _________⎞ ⎪ ⎜ ╱ 2 2 ⎟ ⎪ ⎜ ⎛x⎞ ╱ a - x ⎟ ⎪ a⋅⎜a⋅asin⎜─⎟ - x⋅ ╱ ─────── ⎟ ⎪ ⎜ ⎝a⎠ ╱ 2 ⎟ ⎪ ⎝ ╲╱ a ⎠ ⎪ ───────────────────────────────── otherwise ⎪ 2 ⎩ ⎛ ________ ⎞ ⎜ ╱ 2 ⎟ ⎜ ╱ x ⎟ ⎜ 2 ⎛x⎞ a⋅x⋅ ╱ 1 - ── ⎟ ⎜a ⋅asin⎜─⎟ ╱ 2 _________⎟ ∂ ⎜ ⎝a⎠ ╲╱ a ╱ 2 2 ⎟ ──⎜────────── + ────────────────── - x⋅╲╱ a - x ⎟ ∂x⎝ 2 2 ⎠ ________ ╱ 2 _________ 2 ╱ x ╱ 2 2 2 - a + a⋅ ╱ 1 - ── ⋅╲╱ a - x + 2⋅x ╱ 2 ╲╱ a ─────────────────────────────────────────── _________ ╱ 2 2 ╲╱ a - x ________ ╱ 2 _________ ╱ x ╱ 2 2 a⋅ ╱ 1 - ── - ╲╱ a - x ╱ 2 ╲╱ a$


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="0.001" 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="sample17.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) => x ** 2 / Math.sqrt(2 ** 2 - x ** 2),
g = (x) => -x * Math.sqrt(2 ** 2 - x ** 2) +
x * Math.cos(Math.asin(x / 2)) + 2 ** 2  / 2 * Math.asin(x / 2),
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();