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4.を取り組んでみる。
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m+n≠0I(m,n)=sinm+1xcosn−1m+n+n−1m+nI(m,n−2)I(m,n)=−sinm−1xcosn+1xm+n+m−1m+nI(m−2,n)
(m+n+2)I(m,n+2)=sinm+1xcosn+1x+(n+1)I(m,n)(n+1)I(m,n)=−sinm+1xcosn+1x+(m+n+2)I(m,n+2)n≠−1I(m,n)=−sinm+1xcosn+1xn+1+m+n+2n+1I(m,n+2)I(m+2,n)=−sinm+1xcosn+1xm+n+2+m+1m+n+2I(m,n)(m+1)I(m,n)=sinm+1xcosn+1x+(m+n+2)I(m+2,n)m≠−1I(m,n)=sinm+1xcosn+1xm+1+m+n+2m+1I(m+2,n)
(m+1)I(m,n)=(m+1)∫sinmxcosnxdx=∫(m+1)sinmxcosnxdx=∫(m+1)sinmxcosxcosn−1xdx=∫ddx(sinm+1x)cosn−1xdx=sinm+1xcosn−1x−∫sinm+1x(n−1)cosn−2x(−sinx)dx=sinm+1xcosn−1+(n−1)∫sinm+2xcosn−2xdx=sinm+1xcosn−1+(n−1)∫sinmxcosn−2xsin2xdx=sinm+1xcosn−1+(n−1)∫sinmxcosn−2x(1−cos2x)xdx=sinm+1xcosn−1+(n−1)(∫sinmxcosn−2xdx−∫sinmxcosnxdx)=sinm+1xcosn−1+(n−1)(I(m,n−2)−I(m,n))(m+1)I(m,n)=sinm+1xcosn−1+(n−1)(I(m,n−2)−I(m,n))(m+n)I(m,n)=sinm+1xcosn−1+(n−1)I(m,n−2)(n+1)I(m,n)=∫(n+1)sinmxcosnxdx=∫sinm−1xddx(−cosn+1x)dx=−sinm−1cosn+1x+∫(m−1)sinm−2xcosn+2xdx=−sinm−1xcosn+1x+(m−1)∫sinm−2xcosnxcos2xdx=−sinm−1xcosn+1x+(m−1)∫sinm−2xcosnx(1−sin2x)dx=−sinm−1xcosn+1x+(m−1)(I(m−2,n)−I(m,n))(n+1)I(m,n)=−sinm−1xcosn+1x+(m−1)(I(m−2,n)−I(m,n))(m+n)I(m,n)=−sinm−1xcosn+1x+(m−1)I(m−2,n)
コード(Emacs)
Python 3
#!/usr/bin/env python3 # -*- coding: utf-8 -*- from sympy import pprint, symbols, Integral, sin, cos, plot print('4.') m, n = symbols('m n', integer=True) x = symbols('x') I = Integral(sin(x) ** m + cos(x) ** n, x) pprint(I) m0 = 2 n0 = 3 eq = (I.subs({m: m0, n: n0}) - ( sin(x) ** (m + 1) * cos(x) ** (n - 1) / (m + n) + (n - 1) / (m + n) + I.subs( {m: m0, n: n0 - 2}))).subs({m: m0, n: n0}) pprint(eq)
入出力結果(Terminal, IPython)
$ ./sample4.py 4. ⌠ ⎮ ⎛ m n ⎞ ⎮ ⎝sin (x) + cos (x)⎠ dx ⌡ 3 2 ⌠ ⌠ sin (x)⋅cos (x) ⎮ ⎛ 2 ⎞ ⎮ ⎛ 2 3 ⎞ 2 - ─────────────── - ⎮ ⎝sin (x) + cos(x)⎠ dx + ⎮ ⎝sin (x) + cos (x)⎠ dx - ─ 5 ⌡ ⌡ 5 $
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="m0">m0 = </label> <input id="m0" type="number" step="1" value="2"> <label for="n0">n0 = </label> <input id="n0" type="number" step="1" 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="sample4.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_m0 = document.querySelector('#m0'), input_n0 = document.querySelector('#n0'), inputs = [input_r, input_dx, input_x1, input_x2, input_y1, input_y2, input_m0, input_n0], 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), m0 = parseFloat(input_m0.value), n0 = parseFloat(input_n0.value); if (r === 0 || dx === 0 || x1 > x2 || y1 > y2) { return; } let points = [], lines = [], f = (x) => Math.sin(x) ** m0 * Math.cos(x) ** n0, 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();
(x) => Math.sin(x) ** m0 * Math.cos(x) ** n0,green
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