2017年7月8日土曜日


解析入門〈2〉

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)、8.1(不定積分の計算)、問題1.を取り組んでみる。


    1. 1 x( x 2 −1 ) = A x + B x+1 + C x−1 1 x( x 2 −1 ) = A x 2 −A+B x 2 −Bx+C x 2 +Cx x( x 2 −1 ) 1 x( x 2 −1 ) = ( A+B+C ) x 2 +( −B+C )x−A x( x 2 −1 ) −A=1 A=−1 −1+B+C=0 C=1−B −B+1−B=0 B= 1 2 C= 1 2 ∫ 1 x( x 2 −1 ) =− ∫ 1 x dx+ 1 2 ∫ 1 x+1 dx+ 1 2 ∫ 1 x−1 dx =−log| x |+ 1 2 log| x+1 |+ 1 2 log| x−1 | = 1 2 ( −log | x | 2 +log| x+1 |+log| x−1 | ) = 1 2 log | x 2 −1 | x 2

    2. x 3 −7x+6+7x−6 x 3 −7x+6 =1+ 7x−6 ( x−1 )( x 2 +x−6 ) =1+ 7x−6 ( x−1 )( x−2 )( x+3 ) A x−1 + B x−2 + C x+3 = A x 2 +Ax−6A+B x 2 +2Bx−3B+C x 2 −3Cx+2C ( x−1 )( x−2 )( x+3 ) = ( A+B+C ) x 2 +( A+2B−3C )x−6A−3B+2C ( x−1 )( x−2 )( x+3 ) A+B+C=0 A+2B−3C=7 −6A−3B+2C=−6 C=−A−B A+2B+3A+3B=7 4A+5B=7 −6A−3B−2A−2B=−6 −8A−5B=−6 −4A=1 A=− 1 4 2−5B=−6 B= 8 5 C= 1 4 − 8 5 = 5−32 20 =− 27 20 ∫ 1dx − 1 4 ∫ 1 x−1 dx + 8 5 ∫ 1 x−2 dx − 27 20 ∫ 1 x−3 dx =x− 1 4 log| x−1 |+ 8 5 log| x−2 |− 27 20 log| x−3 |

    3. 1 ( x 2 +1 )( x−1 )( x+1 ) A x−1 + B x+1 + Cx+D x 2 +1 = A x 3 +A x 2 +Ax+A+B x 3 −B x 2 +Bx−B+C x 3 −Cx+D x 2 −D x 4 −1 = ( A+B+C ) x 3 +( A−B+D ) x 2 +( A+B−C )x+A−B−D x 4 −1 A+B+C=0 A−B+D=0 A+B−C=0 A−B−D=1 C=−A−B A+B+A+B=0 A+B=0 B=−A C=−A+A=0 D=−A+B=−A−A=−2A A+A+2A=1 A= 1 4 B=− 1 4 D=− 1 2 1 4 ∫ 1 x−1 dx − 1 4 ∫ 1 x+1 dx− 1 2 ∫ 1 x 2 +1 dx = 1 4 log| x−1 |− 1 4 log| x+1 |− 1 2 arctanx = 1 4 log| x−1 x+1 |−2arctanx

    4. A x + B x+1 + C ( x+1 ) 2 + D ( x+1 ) 3 = A ( x+1 ) 3 +Bx ( x+1 ) 2 +( x 2 +x )C+Dx x ( x+1 ) 3 = A x 3 +3A x 2 +3Ax+A+B x 3 +2B x 2 +Bx+C x 2 +Cx+Dx x ( x+1 ) 3 A+B=1 3A+2B+C=0 3A+B+C+D=0 A=−1 B=2 C=3−4=−1 −3+2−1+D=0 D=2 A x + B x+1 + C ( x+1 ) 2 + D ( x+1 ) 3 − ∫ 1 x dx +2 ∫ 1 x+1 dx − ∫ 1 ( x+1 ) 2 dx +2 ∫ 1 ( x+1 ) 3 dx =−log| x |+2log| x+1 |+ 1 x+1 − 1 ( x+1 ) 2

    5. 1 ( x+1 )( x 2 −x+1 ) A x+1 + Bx+C x 2 −x+1 = A x 2 −Ax+A+B x 2 +Bx+Cx+C ( x+1 )( x 2 −x+1 ) A+B=0 −A+B+C=0 A+C=1 B=−A −A−A+C=0 C=2A A+2A=1 A= 1 3 B=− 1 3 C= 2 3 A x+1 + Bx+C x 2 −x+1 1 3 ∫ 1 x+1 dx − 1 3 ∫ x−2 x 2 −x+1 dx = 1 3 ∫ 1 x+1 dx − 1 3 · 1 2 ∫ 2x−1−3 x 2 −x+1 dx = 1 3 ∫ 1 x+1 dx − 1 3 · 1 2 ∫ 2x−1 x 2 −x+1 dx+ 1 2 ∫ 1 ( x− 1 2 ) 2 + ( 3 2 ) 2 dx ∫ 1 x 3 +1 dx= 1 3 log| x+1 |− 1 6 log( x 2 −x+1 ) + 1 2 · 1 3 2 arctan x− 1 2 3 2 = 1 6 log ( x+1 ) 2 x 2 −x+1 + 1 3 arctan 2x−1 3

    6. ∫ 1 ( x 3 +1 ) 2 dx = ∫ − x 3 + x 3 +1 ( x 3 +1 ) 2 dx = ∫ − x 3 ( x 3 +1 ) 2 dx + ∫ 1 x 3 +1 dx = 1 3 ∫ x d dx ( 1 x 3 +1 )dx + ∫ 1 x 3 +1 dx = 1 3 ( x x 3 +1 − ∫ 1 x 3 +1 dx )+ ∫ 1 x 3 +1 dx = x 3( x 3 +1 ) + 2 3 ∫ 1 x 3 +1 dx = x 3( x 3 +1 ) + 2 3 ( 1 6 log ( x+1 ) 2 x 2 −x+1 + 1 3 arctan 2x−1 3 ) = x 3( x 3 +1 ) + 1 9 log ( x+1 ) 2 x 2 −x+1 + 2 3 3 arctan 2x−1 3

コード(Emacs)

Python 3

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

from sympy import pprint, symbols, Integral
import random

x = symbols('x')
fs = [1 / (x ** 3 - x),
      x ** 3 / (x ** 3 - 7 * x + 6),
      1 / (x ** 4 - 1),
      (x ** 3 - 1) / (x * (x + 1) ** 3),
      1 / (x ** 3 + 1),
      1 / (x ** 3 + 1) ** 2]

for i, f in enumerate(fs, 1):
    print(f'({i})')
    I = Integral(f, x)
    pprint(I)
    pprint(I.doit())
    print()

入出力結果(Terminal, IPython)

$ ./sample1.py
(1)
⌠          
⎮   1      
⎮ ────── dx
⎮  3       
⎮ x  - x   
⌡          
             ⎛ 2    ⎞
          log⎝x  - 1⎠
-log(x) + ───────────
               2     

(2)
⌠                
⎮       3        
⎮      x         
⎮ ──────────── dx
⎮  3             
⎮ x  - 7⋅x + 6   
⌡                
    8⋅log(x - 2)   log(x - 1)   27⋅log(x + 3)
x + ──────────── - ────────── - ─────────────
         5             4              20     

(3)
⌠          
⎮   1      
⎮ ────── dx
⎮  4       
⎮ x  - 1   
⌡          
log(x - 1)   log(x + 1)   atan(x)
────────── - ────────── - ───────
    4            4           2   

(4)
⌠              
⎮    3         
⎮   x  - 1     
⎮ ────────── dx
⎮          3   
⎮ x⋅(x + 1)    
⌡              
     x                              
──────────── - log(x) + 2⋅log(x + 1)
 2                                  
x  + 2⋅x + 1                        

(5)
⌠          
⎮   1      
⎮ ────── dx
⎮  3       
⎮ x  + 1   
⌡          
                                      ⎛2⋅√3⋅x   √3⎞
                ⎛ 2        ⎞   √3⋅atan⎜────── - ──⎟
log(x + 1)   log⎝x  - x + 1⎠          ⎝  3      3 ⎠
────────── - ─────────────── + ────────────────────
    3               6                   3          

(6)
⌠             
⎮     1       
⎮ ───────── dx
⎮         2   
⎮ ⎛ 3    ⎞    
⎮ ⎝x  + 1⎠    
⌡             
                                                     ⎛2⋅√3⋅x   √3⎞
                             ⎛ 2        ⎞   2⋅√3⋅atan⎜────── - ──⎟
   x       2⋅log(x + 1)   log⎝x  - x + 1⎠            ⎝  3      3 ⎠
──────── + ──────────── - ─────────────── + ──────────────────────
   3            9                9                    9           
3⋅x  + 3                                                          

$

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.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 f5 = (x) => 1 / (x ** 3 + 1),
    f6 = (x) => 1 / (x ** 3 + 1) ** 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 = [],
        fns = [[f5, 'green'], [f6, 'orange']];

    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();




































						












時刻:


12:00




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