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2017年7月13日木曜日

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)、8.1(不定積分の計算)、問題7.を取り組んでみる。


  1. 2tanx21+tan2x2=2sinx2cosx21+sin2x2cos2x2=2sinx2cosx2=sinx1tan2x21+tan2x2=1sin2x2cos2x21+sin2x2cos2x2=cos2x2sin2x2=cosxdxdt=1dtdx=11cos2x2·12=2sin2x2+cos2x2cos2x2=21+tan2x2=21+t2

    1. 11+1t21+t2·21+t2dt=21+t2+1t2·dt=1dt=t=tanx2

    2. 11t21+t2·21+t2dt=21t2dt=2(1+t)(1t)dtA1+t+B1t=(BA)t+A+B(1+t)(1t)BA=0A+B=2A=B2A=2A=1B=12(1+t)(1t)dt=11+tdt+11tdt=log|1+t|log|1t|=log|1+t1t|=log|1+tanx21tanx2|

    3. 2t1+t21+2t1+t2·21+t2dt=2t1+t2+2t·21+t2dt=4t(t+1)2(t2+1)dtAt+1+B(t+1)2+Ct+Dt2+1=A(t+1)(t2+1)+B(t2+1)+(Ct+D)(t+1)2(t+1)2(t2+1)A(t+1)(t2+1)+B(t2+1)+(Ct+D)(t+1)2=A(t3+t2+t+1)+Bt2+B+(Ct+D)(t2+2t+1)=(A+C)t3+(A+B+2C+D)t2+(A+C+2D)t+A+B+DA+C=0A+B+2C+D=0A+C+2D=1A+B+D=0C=AA+B2A+D=0AA+2D=1A+B+D=0D=ABA+B2AAB=0A=0D=B2D=1D=12B=12C=04t(t+1)2(t2+1)dt=4(12(t+1)2+12t2+1)dt=2(1(t+1)2+1t2+1)dt=2((t+1)1+arctant)=2(1tanx2+1+arctan(tanx2))=2tanx2+1+2x2=2tanx2+1+x

コード(Emacs)

Python 3

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

from sympy import pprint, symbols, Integral, sin, cos

print('7.')
x = symbols('x')

fs = [1 / (1 + cos(x)),
      1 / cos(x),
      sin(x) / (1 + sin(x))]


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

入出力結果(Terminal, IPython)

$ ./sample7.py
7.
(1)
⌠              
⎮     1        
⎮ ────────── dx
⎮ cos(x) + 1   
⌡              
   ⎛x⎞
tan⎜─⎟
   ⎝2⎠
factor:
   ⎛x⎞
tan⎜─⎟
   ⎝2⎠
expand:
   ⎛x⎞
tan⎜─⎟
   ⎝2⎠
(2)
⌠          
⎮   1      
⎮ ────── dx
⎮ cos(x)   
⌡          
  log(sin(x) - 1)   log(sin(x) + 1)
- ─────────────── + ───────────────
         2                 2       
factor:
-log(sin(x) - 1) + log(sin(x) + 1)
──────────────────────────────────
                2                 
expand:
  log(sin(x) - 1)   log(sin(x) + 1)
- ─────────────── + ───────────────
         2                 2       
(3)
⌠              
⎮   sin(x)     
⎮ ────────── dx
⎮ sin(x) + 1   
⌡              
      ⎛x⎞                           
 x⋅tan⎜─⎟                           
      ⎝2⎠        x            2     
────────── + ────────── + ──────────
   ⎛x⎞          ⎛x⎞          ⎛x⎞    
tan⎜─⎟ + 1   tan⎜─⎟ + 1   tan⎜─⎟ + 1
   ⎝2⎠          ⎝2⎠          ⎝2⎠    
factor:
     ⎛x⎞        
x⋅tan⎜─⎟ + x + 2
     ⎝2⎠        
────────────────
      ⎛x⎞       
   tan⎜─⎟ + 1   
      ⎝2⎠       
expand:
      ⎛x⎞                           
 x⋅tan⎜─⎟                           
      ⎝2⎠        x            2     
────────── + ────────── + ──────────
   ⎛x⎞          ⎛x⎞          ⎛x⎞    
tan⎜─⎟ + 1   tan⎜─⎟ + 1   tan⎜─⎟ + 1
   ⎝2⎠          ⎝2⎠          ⎝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="-10">
<label for="x2">x2 = </label>
<input id="x2" type="number" value="10">
<br>
<label for="y1">y1 = </label>
<input id="y1" type="number" value="-10">
<label for="y2">y2 = </label>
<input id="y2" type="number" value="10">

<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="sample7.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 f1 = (x) => 1 / (1 + Math.cos(x)),
    f2 = (x) => 1 / Math.cos(x),
    f3 = (x) => Math.sin(x) / (1 + Math.sin(x));

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 = [[f1, 'red'],
               [f2, 'green'],
               [f3, 'blue']];

    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();
-10-8-6-4-20246810-10-8-6-4-20246810
(x) => 1 / (1 + Math.cos(x)),red
(x) => 1 / Math.cos(x),green
(x) => Math.sin(x) / (1 + Math.sin(x)),blue


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