数学 - Python - JavaScript - 解析学 - 積分法 - 定積分の計算(部分積分法、置換積分法、三角関数(正弦、余弦、逆正弦))

解析入門〈2〉

学習環境

• Surface 3 (4G LTE)、Surface 3 タイプ カバー、Surface ペン(端末)
• Windows 10 Pro (OS)
• 数式入力ソフト(TeX, MathML): MathType
• MathML対応ブラウザ: Firefox、Safari
• MathML非対応ブラウザ(Internet Explorer, Microsoft Edge, Google Chrome...)用JavaScript Library: MathJax
• 参考書籍
  • Pythonからはじめる数学入門
  • JavaScriptリファレンス 第6版
  • インタラクティブ・データビジュアライゼーション

解析入門〈2〉(松坂 和夫(著)、岩波書店)の第8章(積分の計算)、8.2(定積分の計算)、問題1-6.を取り組んでみる。

1. 6. 
      

      部分積分法。

      $\begin{array}{l}{\displaystyle \int \frac{x\arcsin x}{\sqrt{1-x^2}}dx}\\ =x\left(\arcsin x\right)\left(\arcsin x\right)-{\displaystyle \int \left(\arcsin x+x\frac{1}{\sqrt{1-x^2}}\right)\arcsin xdx}\\ =x{\left(\arcsin x\right)}^2-{\displaystyle \int {\left(\arcsin x\right)}^{2}dx}-{\displaystyle \int \frac{x\arcsin x}{\sqrt{1-x^2}}dx}\end{array}$

      置換積分法。

      $\begin{array}{l}t=\arcsin x\\ x=-1,t=-\frac{\pi }{2}\\ x=1,t=\frac{\pi }{2}\\ \sin t=x\\ \frac{dx}{dt}=\cos t\\ \\ x{\left(\arcsin x\right)}^2-{\displaystyle \int {\left(\arcsin x\right)}^{2}dx}-{\displaystyle \int \frac{x\arcsin x}{\sqrt{1-x^2}}dx}\\ =x{\left(\arcsin x\right)}^2-{\displaystyle \int t^2\cos tdt}-{\displaystyle \int \frac{x\arcsin x}{\sqrt{1-x^2}}dx}\end{array}$

      部分積分法。

      $\begin{array}{l}{\displaystyle \int t^2\cos tdt}\\ =t^2\sin t-{\displaystyle \int 2t\sin tdt}\\ =t^2\sin t-2\left(-t\cos t+{\displaystyle \int \cos tdt}\right)\\ \\ x{\left(\arcsin x\right)}^2-{\displaystyle \int t^2\cos tdt}-{\displaystyle \int \frac{x\arcsin x}{\sqrt{1-x^2}}dx}\\ =x{\left(\arcsin x\right)}^2-t^2\sin t-2t\cos t+2{\displaystyle \int \cos tdt}-{\displaystyle \int \frac{x\arcsin x}{\sqrt{1-x^2}}dx}\\ =x{\left(\arcsin x\right)}^2-t^2\sin t-2t\cos t+2\sin t-{\displaystyle \int \frac{x\arcsin x}{\sqrt{1-x^2}}dx}\\ \\ {\displaystyle \int \frac{x\arcsin x}{\sqrt{1-x^2}}dx}=\frac{1}{2}\left(x{\left(\arcsin x\right)}^2-t^2\sin t-2t\cos t+2\sin t\right)\\ \\ {\displaystyle {\int }_{-1}^1\frac{x\arcsin x}{\sqrt{1-x^2}}}\\ =\frac{1}{2}\left({\left[x{\left(\arcsin x\right)}^2\right]}_{-1}^1+{\left[-t^2\sin t-2t\cos t+2\sin t\right]}_{-\frac{\pi }{2}}^{\frac{\pi }{2}}\right)\\ =\frac{1}{2}\left({\left(\arcsin 1\right)}^2+{\left(\arcsin \left(-1\right)\right)}^2+\left(-\frac{{\pi }^2}{4}+2\right)-\left(\frac{{\pi }^2}{4}-2\right)\right)\\ =\frac{1}{2}\left(\frac{{\pi }^2}{4}+\frac{{\pi }^2}{4}-\frac{{\pi }^2}{4}+2-\frac{{\pi }^2}{4}+2\right)\\ =2\end{array}$

コード(Emacs)

Python 3

#!/usr/bin/env python3
# -*- coding: utf-8 -*-

from sympy import pprint, symbols, Integral, sqrt, asin, plot

print('1.')
x = symbols('x', real=True)
a = symbols('a', positive=True)
fs = [(x * asin(x) / sqrt(1 - x ** 2), (-1, 1))]

for i, (f, (x1, x2)) in enumerate(fs, 6):
    print(f'({i})')
    I = Integral(f, (x, x1, x2))
    pprint(I)
    I = I.doit()
    pprint(I)
    print('factor:')
    pprint(I.factor())
    print('expand:')
    pprint(I.expand())

p = plot(fs[0][0], show=False, legend=True)
p.save('sample1_6.svg')

入出力結果(Terminal, IPython)

$ ./sample1_6.py
1.
(6)
1                  
⌠                  
⎮    x⋅asin(x)     
⎮  ───────────── dx
⎮     __________   
⎮    ╱    2        
⎮  ╲╱  - x  + 1    
⌡                  
-1                 
2
factor:
2
expand:
2
$

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="0.0001" 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_6.js"></script>    

JavaScript

let div0 = document.querySelector('#graph0'),
    pre0 = document.querySelector('#output0'),
    width = 600,
    height = 600,
    padding = 50,
    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 f = (x) => x * Math.asin(x) / Math.sqrt(1 - x ** 2);

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 = [[-1, y1, -1, y2, 'red'],
                 [1, y1, 1, y2, 'red']],
        fns = [[f, 'green']];

    fns
        .forEach((o) => {
            let [fn, color] = o;
            
            for (let x = x1; x <= x2; x += dx) {
                let y = fn(x);
                
                if (Math.abs(y) < Infinity) {
                    points.push([x, y, color]);
                }
            }
        });
    
    let xscale = d3.scaleLinear()
        .domain([x1, x2])
        .range([padding, width - padding]);

    let yscale = d3.scaleLinear()
        .domain([y1, y2])
        .range([height - padding, padding]);

    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('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.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.append('g')
        .attr('transform', `translate(0, ${height - padding})`)
        .call(xaxis);

    svg.append('g')
        .attr('transform', `translate(${padding}, 0)`)
        .call(yaxis);
    p(fns.join('\n'));
};

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

r = dx =
x1 = x2 =
y1 = y2 =