## 2018年5月25日金曜日

### 数学 - Python - JavaScript - 解析学 - 積分 - 積分の計算 - 三角関数の積分(正弦、累乗(べき乗)、倍角)

1. 定理1より。

$\begin{array}{}\int {\mathrm{sin}}^{4}x\mathrm{dx}\\ =-\frac{1}{4}{\mathrm{sin}}^{3}x\mathrm{cos}x+\frac{3}{4}\int {\mathrm{sin}}^{2}x\mathrm{dx}\\ =-\frac{1}{4}{\mathrm{sin}}^{3}x\mathrm{cos}x+\frac{3}{4}\left(-\frac{1}{2}\mathrm{sin}x\mathrm{cos}x+\frac{1}{2}\int 1\mathrm{dx}\right)\\ =-\frac{1}{4}{\mathrm{sin}}^{3}x\mathrm{cos}x-\frac{3}{8}\mathrm{sin}x\mathrm{cos}x+\frac{3}{8}x\end{array}$

1つずつ計算していった場合。

$\begin{array}{}\int {\mathrm{sin}}^{4}x\mathrm{dx}\\ =-\mathrm{cos}x{\mathrm{sin}}^{3}x+\int 3{\mathrm{cos}}^{2}x{\mathrm{sin}}^{2}x\mathrm{dx}\\ =-\mathrm{cos}x{\mathrm{sin}}^{3}x+3\int \left(1-{\mathrm{sin}}^{2}x\right){\mathrm{sin}}^{2}x\mathrm{dx}\\ =-\mathrm{cos}x{\mathrm{sin}}^{3}x+3\int {\mathrm{sin}}^{2}x\mathrm{dx}-3\int {\mathrm{sin}}^{4}x\mathrm{dx}\\ 4\int {\mathrm{sin}}^{4}x\mathrm{dx}\\ =-\mathrm{cos}x{\mathrm{sin}}^{3}x+3\int \frac{1-\mathrm{cos}2x}{2}\mathrm{dx}\\ =-\mathrm{cos}x{\mathrm{sin}}^{3}x+\frac{3}{2}\left(x-\frac{\mathrm{sin}2x}{2}\right)\\ =-\mathrm{cos}x{\mathrm{sin}}^{3}x+\frac{3}{2}x-\frac{3}{2}\mathrm{sin}x\mathrm{cos}x\\ \int {\mathrm{sin}}^{4}x\mathrm{dx}=-\frac{1}{.4}{\mathrm{sin}}^{3}x\mathrm{cos}x-\frac{3}{8}\mathrm{sin}x\mathrm{cos}x+\frac{3}{8}x\end{array}$

コード(Emacs)

Python 3

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

x = symbols('x')
f = sin(x) ** 4
I = Integral(f)
for t in [I, I.doit()]:
pprint(t)
print()

p = plot(*[sin(x) ** i for i in range(1, 5)], legend=True, show=False)

for i, color in enumerate(['red', 'green', 'blue', 'orange']):
p[i].line_color = color

p.save('sample1.svg')


$./sample1.py ⌠ ⎮ 4 ⎮ sin (x) dx ⌡ 3 3⋅x sin (x)⋅cos(x) 3⋅sin(x)⋅cos(x) ─── - ────────────── - ─────────────── 8 4 8$


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="sample1.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'),
inputs = [input_r, input_dx, input_x1, input_x2, input_y1, input_y2],
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 fns = [[(x) => Math.sin(x), 'red'],
[(x) => Math.sin(x) ** 2, 'green'],
[(x) => Math.sin(x) ** 3, 'blue'],
[(x) => Math.sin(x) ** 4, 'orange']];

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 = [],
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();