数学 - Python - 基本理論(Basic Theory) - 積分(Integration) - 積分法の基礎公式 - 置換積分法, 部分積分法(定積分を求める)

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

オイラーの贈物―人類の至宝eiπ=-1を学ぶ (吉田武(著)、東海大学出版会)の第Ⅰ部(基礎理論(Basic Theory))、第4章(積分(Integration))、4.4(積分法の基礎公式)、4.4.1(置換積分法)、4.4.2(部分積分法)、問題3.を取り組んでみる。

1. $\begin{array}{l} a x + b = t \\ a = \frac{d t}{d x} \\ a d x = d t \\ x = 0 \\ t = b \\ x = 1 \\ t = a + b \\ \int_{b}^{a + b} \frac{1}{2 \sqrt{t}} d t \\ = {[\sqrt{t}]}_{b}^{a + b} \\ = \sqrt{a + b} - \sqrt{b} \end{array}$
2. $\begin{array}{l} \int_{0}^{1} \frac{x}{\sqrt{a x + b}} d x \\ = {[\frac{2}{a} x \sqrt{a x + b}]}_{0}^{1} - \int_{0}^{1} \frac{2}{a} \sqrt{a x + b} d x \\ = \frac{2}{a} \sqrt{a + b} - {[\frac{2}{a} \frac{2}{3} \frac{1}{a} {(a x + b)}^{\frac{3}{2}}]}_{0}^{1} \\ = \frac{2 \sqrt{a + b}}{a} - \frac{4}{3 a^{2}} ({(a + b)}^{\frac{3}{2}} - b^{\frac{3}{2}}) \end{array}$

コード(Emacs)

Python 3

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

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

print('2.')
x, a, b = symbols('x a b')
fs = [(a / (2 * sqrt(a * x + b)), (0, 1)),
      (x / sqrt(a * x + b), (0, 1))]


for i, (f, (x1, x2)) in enumerate(fs, 1):
    print(f'[{i}]')
    try:
        I = Integral(f, (x, x1, x2))
        pprint(I)
        pprint(I.doit())
        p = plot(f.subs({a: 2, b: 3}), show=False, legend=True)
        p.save(f'sample3_{i}.svg')
    except Exception as err:
        print(type(err), err)
    print()

入出力結果(Terminal, IPython)

$ ./sample3.py
2.
[1]
1                 
⌠                 
⎮       a         
⎮ ───────────── dx
⎮     _________   
⎮ 2⋅╲╱ a⋅x + b    
⌡                 
0                 
        _______
-√b + ╲╱ a + b 

[2]
1               
⌠               
⎮      x        
⎮ ─────────── dx
⎮   _________   
⎮ ╲╱ a⋅x + b    
⌡               
0               
<class 'TypeError'> bad operand type for unary -: 'And'

$

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">
<br>
<label for="a0">a0 = </label>
<input id="a0" type="number" value="2">
<label for="b0">b0 = </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="sample3.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'),
    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 = [[0, y1, 0, y2, 'red'],
                 [1, y1, 1, y2, 'red']],
        f1 = (x) => a0 / (2 * Math.sqrt(a0 * x + b0)),
        f2 = (x) => x / Math.sqrt(a0 * x + b0),
        fns = [[f1, 'green'],
               [f2, 'orange']],
        fns1 = [],
        fns2 = [];

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

                if (Math.abs(y) < Infinity) {
                    points.push([x, y, 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])
        .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('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')));
    p(Math.sqrt(a0 + b0) - Math.sqrt(b0));
    p(2 * Math.sqrt(a0 + b0) / a0 -
      4 / (3 * a0 ** 2) * (Math.sqrt((a0 + b0) ** 3) - Math.sqrt(b0 ** 3)));
};

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

r = dx =
x1 = x2 =
y1 = y2 =
a0 = b0 =

Kamimura's blog

2017年8月25日金曜日

数学 - Python - 基本理論(Basic Theory) - 積分(Integration) - 積分法の基礎公式 - 置換積分法, 部分積分法(定積分を求める)

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