## 2018年11月7日水曜日

### 数学 - Python - JavaScript - 解析学 - 積分 - 積分の応用 - 曲線の長さ(指数関数、置換積分法、部分分数分解、対数関数)

1. $\begin{array}{}{\int }_{0}^{1}\sqrt{1+{\left(\frac{d}{\mathrm{dx}}{e}^{x}\right)}^{2}}\mathrm{dx}\\ ={\int }_{0}^{1}\sqrt{1+{e}^{2x}}\mathrm{dx}\\ 1+{e}^{2x}={t}^{2}\\ 2{e}^{2x}=2t\frac{\mathrm{dt}}{\mathrm{dx}}\\ \mathrm{dx}=\frac{t}{{e}^{2t}}=\frac{t}{{t}^{2}-1}\\ {\int }_{\sqrt{2}}^{\sqrt{1+{e}^{2}}}\frac{{t}^{2}}{{t}^{2}-1}\mathrm{dt}\\ =\underset{\sqrt{2}}{\overset{\sqrt{1+{e}^{2}}}{\int }}\left(1+\frac{1}{{t}^{2}-1}\right)\mathrm{dx}\\ \frac{1}{{t}^{2}-1}=\frac{1}{\left(t+1\right)\left(t-1\right)}\\ \frac{a}{t+1}+\frac{b}{t-1}=\frac{\left(a+b\right)t+\left(b-a\right)}{{t}^{2}-1}\\ a+b=0\\ b-a=1\\ b=\frac{1}{2}\\ a=-\frac{1}{2}\\ \sqrt{1+{e}^{2}}-\sqrt{2}+\frac{1}{2}\underset{\sqrt{2}}{\overset{\sqrt{1+{e}^{2}}}{\int }}\left(\frac{1}{t-1}-\frac{1}{t+1}\right)\mathrm{dt}\\ =\sqrt{1+{e}^{2}}-\sqrt{2}+\frac{1}{2}\left(\mathrm{log}\left(\sqrt{1+{e}^{2}}-1\right)-\mathrm{log}\left(\sqrt{2}-1\right)-\mathrm{log}\left(\sqrt{1+{e}^{2}}+1\right)+\mathrm{log}\left(\sqrt{2}+1\right)\right)\\ =\sqrt{1+{e}^{2}}-\sqrt{2}+\frac{1}{2}\mathrm{log}\left(\frac{\sqrt{1+{e}^{2}}-1}{\sqrt{1+{e}^{2}}+1}·\frac{\sqrt{2}+1}{\sqrt{2}-1}\right)\end{array}$

コード(Emacs)

Python 3

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

print('5.')
x = symbols('x')
f = exp(x)
I = Integral(sqrt(1 + Derivative(f, x, 1) ** 2), (x, 0, 1))

for t in [I, I.doit()]:
pprint(t.simplify())
print()

p = plot(f, show=False, legend=True)
p.save('sample5.svg')

I = Integral(x ** 2 / (x ** 2 - 1), (x, sqrt(2), sqrt(1 + exp(2))))
for t in [I, I.doit()]:
pprint(t.simplify())
print()


$./sample5.py 5. 1 ⌠ ⎮ _______________ ⎮ ╱ 2 ⎮ ╱ ⎛d ⎛ x⎞⎞ ⎮ ╱ ⎜──⎝ℯ ⎠⎟ + 1 dx ⎮ ╲╱ ⎝dx ⎠ ⌡ 0 1 ⌠ ⎮ __________ ⎮ ╱ 2⋅x ⎮ ╲╱ ℯ + 1 dx ⌡ 0 ________ ╱ 2 ╲╱ 1 + ℯ ⌠ ⎮ 2 ⎮ x ⎮ ────── dx ⎮ 2 ⎮ x - 1 ⌡ √2 ⎛ ________⎞ ⎛ ________⎞ ⎜ ╱ 2 ⎟ ⎜ ╱ 2 ⎟ log⎝1 + ╲╱ 1 + ℯ ⎠ log⎝-1 + ╲╱ 1 + ℯ ⎠ log(1 + √2) log(-1 + √2 -√2 - ──────────────────── + ───────────────────── + ─────────── - ─────────── 2 2 2 2 ________ ) ╱ 2 ─ + ╲╱ 1 + ℯ$


HTML5

<div id="graph0"></div>
<pre id="output0"></pre>
<label for="r0">r = </label>
<input id="r0" type="number" min="0" value="1">
<label for="dx">dx = </label>
<input id="dx" type="number" min="0" step="0.001" value="0.01">
<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="sample5.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 fns = [[(x) => Math.exp(x), 'red']];

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 = [[0, y1, 0, y2, 'green'],
[1, y1, 1, y2, 'blue']];

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();