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
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。
2tanx21+tan2x2=2sinx2cosx21+sin2x2cos2x2=2sinx2cosx2=sinx1−tan2x21+tan2x2=1−sin2x2cos2x21+sin2x2cos2x2=cos2x2−sin2x2=cosxdxdt=1dtdx=11cos2x2·12=2sin2x2+cos2x2cos2x2=21+tan2x2=21+t2
∫11+1−t21+t2·21+t2dt=∫21+t2+1−t2·dt=∫1dt=t=tanx2
∫11−t21+t2·21+t2dt=∫21−t2dt=∫2(1+t)(1−t)dtA1+t+B1−t=(B−A)t+A+B(1+t)(1−t)B−A=0A+B=2A=B2A=2A=1B=1∫2(1+t)(1−t)dt=∫11+tdt+∫11−tdt=log|1+t|−log|1−t|=log|1+t1−t|=log|1+tanx21−tanx2|
∫2t1+t21+2t1+t2·21+t2dt=∫2t1+t2+2t·21+t2dt=4∫t(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+B−2A+D=0A−A+2D=1A+B+D=0D=−A−BA+B−2A−A−B=0A=0D=−B2D=1D=12B=−12C=04∫t(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();
(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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