2017年7月25日火曜日

学習環境

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. 部分積分法。

      xarcsinx 1 x 2 dx =x( arcsinx )( arcsinx ) ( arcsinx+x 1 1 x 2 )arcsinxdx =x ( arcsinx ) 2 ( arcsinx ) 2 dx xarcsinx 1 x 2 dx

      置換積分法。

      t=arcsinx x=1,t= π 2 x=1,t= π 2 sint=x dx dt =cost x ( arcsinx ) 2 ( arcsinx ) 2 dx xarcsinx 1 x 2 dx =x ( arcsinx ) 2 t 2 costdt xarcsinx 1 x 2 dx

      部分積分法。

      t 2 costdt = t 2 sint 2tsintdt = t 2 sint2( tcost+ costdt ) x ( arcsinx ) 2 t 2 costdt xarcsinx 1 x 2 dx =x ( arcsinx ) 2 t 2 sint2tcost+2 costdt xarcsinx 1 x 2 dx =x ( arcsinx ) 2 t 2 sint2tcost+2sint xarcsinx 1 x 2 dx xarcsinx 1 x 2 dx = 1 2 ( x ( arcsinx ) 2 t 2 sint2tcost+2sint ) 1 1 xarcsinx 1 x 2 = 1 2 ( [ x ( arcsinx ) 2 ] 1 1 + [ t 2 sint2tcost+2sint ] π 2 π 2 ) = 1 2 ( ( arcsin1 ) 2 + ( arcsin( 1 ) ) 2 +( π 2 4 +2 )( π 2 4 2 ) ) = 1 2 ( π 2 4 + π 2 4 π 2 4 +2 π 2 4 +2 ) =2

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();







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