<!DOCTYPE html>
<html lang="en">
<head>
  <meta charset="UTF-8">
  <meta name="viewport" content="width=device-width, initial-scale=1.0">
  <title>Constant Balance Technique</title>
  <style>
    * {
      box-sizing: border-box;
      margin: 0;
      padding: 0;
      font-family: -apple-system, BlinkMacSystemFont, "Segoe UI", Roboto, Helvetica, Arial, sans-serif;
    }

    :root {
      --primary: #22b7ff;
      --primary-dark: #0090d4;
      --primary-soft: #e8f7ff;
      --text: #1e2945;
      --muted: #64748b;
      --border: #e6e6e6;
      --bg: #ffffff;
      --accent-amber: #d97706;
      --accent-amber-bg: #fef3c7;
      --accent-green: #16a34a;
      --accent-green-bg: #dcfce7;
    }

    body {
      background: transparent;
      color: var(--text);
      display: flex;
      flex-direction: column;
      align-items: center;
      justify-content: flex-start;
      padding: 8px 10px;
      min-height: 100vh;
      overflow-x: hidden;
    }

    .diagram-container {
      width: 100%;
      max-width: 520px;
      display: flex;
      flex-direction: column;
      gap: 10px;
    }

    /* Math typography */
    .math {
      font-family: "Cambria Math", "Latin Modern Math", "Times New Roman", Times, serif;
      font-size: 1.05em;
      letter-spacing: 0.3px;
    }
    
    .math var {
      font-style: italic;
      font-family: inherit;
    }

    .math sup {
      font-size: 0.72em;
      vertical-align: 0.55em;
      line-height: 0;
      font-weight: 600;
    }

    .math sub {
      font-size: 0.72em;
      vertical-align: -0.3em;
      line-height: 0;
    }

    .frac {
      display: inline-flex;
      flex-direction: column;
      vertical-align: -0.45em;
      text-align: center;
      padding: 0 2px;
      font-size: 0.9em;
      line-height: 1;
    }

    .num {
      border-bottom: 1.5px solid currentColor;
      padding-bottom: 1px;
      display: block;
    }

    .den {
      padding-top: 1px;
      display: block;
    }

    .big-sym {
      font-size: 1.25em;
      vertical-align: -0.08em;
      margin-right: 1px;
    }

    /* Target Problem Header */
    .problem-strip {
      background: #f8fafc;
      border: 1px solid var(--border);
      border-radius: 10px;
      padding: 8px 12px;
      display: flex;
      align-items: center;
      justify-content: center;
      gap: 8px;
      flex-wrap: wrap;
      text-align: center;
    }

    .problem-label {
      font-size: 11px;
      font-weight: 800;
      color: var(--muted);
      text-transform: uppercase;
      letter-spacing: 0.6px;
      background: #e2e8f0;
      padding: 3px 7px;
      border-radius: 5px;
    }

    .problem-math {
      font-size: 1.15em;
      font-weight: 600;
      color: var(--text);
    }

    /* Stepper Navigation */
    .step-nav {
      display: flex;
      gap: 6px;
      width: 100%;
      justify-content: space-between;
    }

    .nav-btn {
      flex: 1;
      padding: 8px 4px;
      font-size: 12px;
      font-weight: 700;
      border-radius: 8px;
      border: 1px solid var(--border);
      background: #ffffff;
      color: var(--muted);
      cursor: pointer;
      display: flex;
      align-items: center;
      justify-content: center;
      gap: 5px;
      transition: background-color 0.2s, border-color 0.2s, color 0.2s;
    }

    .nav-btn .num-badge {
      display: inline-flex;
      align-items: center;
      justify-content: center;
      width: 18px;
      height: 18px;
      border-radius: 50%;
      background: #e2e8f0;
      color: #475569;
      font-size: 11px;
    }

    .nav-btn.active {
      background: var(--primary-soft);
      border-color: var(--primary);
      color: var(--primary-dark);
    }

    .nav-btn.active .num-badge {
      background: var(--primary);
      color: #ffffff;
    }

    /* Step Card Stage */
    .stage {
      position: relative;
      width: 100%;
      min-height: 210px;
      height: 215px;
      background: #ffffff;
      border: 1px solid var(--border);
      border-radius: 12px;
      overflow: hidden;
    }

    .step-panel {
      position: absolute;
      inset: 0;
      padding: 12px 14px;
      display: flex;
      flex-direction: column;
      align-items: center;
      justify-content: space-between;
      opacity: 0;
      pointer-events: none;
      transition: opacity 0.25s ease;
    }

    .step-panel.active {
      opacity: 1;
      pointer-events: auto;
    }

    /* Cards inside Step 1 & 3 */
    .decompose-row {
      display: flex;
      align-items: stretch;
      justify-content: center;
      gap: 8px;
      width: 100%;
    }

    .badge-card {
      flex: 1;
      background: #f8fafc;
      border: 1px solid var(--border);
      border-radius: 8px;
      padding: 8px 10px;
      text-align: center;
      display: flex;
      flex-direction: column;
      align-items: center;
      justify-content: center;
      gap: 6px;
    }

    .badge-tag {
      font-size: 10px;
      font-weight: 800;
      text-transform: uppercase;
      padding: 2px 7px;
      border-radius: 10px;
      letter-spacing: 0.5px;
      white-space: nowrap;
    }

    .tag-blue { background: #e0f2fe; color: #0284c7; }
    .tag-amber { background: var(--accent-amber-bg); color: var(--accent-amber); }
    .tag-green { background: var(--accent-green-bg); color: var(--accent-green); }

    .card-formula {
      font-size: 1.05em;
      color: var(--text);
      font-weight: bold;
    }

    /* Step 2 Balance Visual */
    .balance-container {
      display: flex;
      flex-direction: column;
      align-items: center;
      width: 100%;
      gap: 6px;
    }

    .balance-visual {
      display: flex;
      align-items: center;
      justify-content: center;
      gap: 8px;
      background: #f0f9ff;
      border: 1px dashed var(--primary);
      border-radius: 10px;
      padding: 6px 12px;
      width: 100%;
      max-width: 380px;
    }

    .factor-box {
      background: #ffffff;
      border: 1px solid var(--primary);
      border-radius: 6px;
      padding: 4px 8px;
      font-weight: 700;
      font-size: 0.95em;
      display: flex;
      align-items: center;
      gap: 4px;
    }

    .factor-desc {
      font-size: 10px;
      color: var(--muted);
      font-weight: 600;
      text-transform: uppercase;
    }

    .multiply-sym {
      color: var(--primary-dark);
      font-weight: bold;
      font-size: 16px;
    }

    .neutral-pill {
      font-size: 11px;
      font-weight: 800;
      color: #0369a1;
      background: #e0f2fe;
      padding: 4px 8px;
      border-radius: 10px;
      white-space: nowrap;
    }

    /* Formula Result Strip */
    .formula-box {
      width: 100%;
      background: #f8fafc;
      border: 1px solid var(--border);
      border-radius: 8px;
      padding: 8px 10px;
      display: flex;
      align-items: center;
      justify-content: center;
      text-align: center;
      font-size: 13.5px;
      min-height: 42px;
    }

    .success-box {
      background: #f0fdf4;
      border-color: #bbf7d0;
      color: #14532d;
    }

    /* Controls */
    .controls {
      display: flex;
      align-items: center;
      justify-content: space-between;
      width: 100%;
      padding: 0 2px;
    }

    .btn-ctrl {
      background: #ffffff;
      border: 1px solid var(--border);
      border-radius: 8px;
      padding: 6px 14px;
      font-size: 12px;
      font-weight: 700;
      color: var(--text);
      cursor: pointer;
      display: flex;
      align-items: center;
      gap: 4px;
      transition: background-color 0.15s, border-color 0.15s;
    }

    .btn-ctrl:hover {
      background: #f8fafc;
      border-color: #cbd5e1;
    }

    .dots-wrap {
      display: flex;
      gap: 6px;
      align-items: center;
    }

    .dot {
      width: 8px;
      height: 8px;
      border-radius: 50%;
      background: var(--border);
      cursor: pointer;
      transition: background-color 0.2s, width 0.2s;
    }

    .dot.active {
      background: var(--primary);
      width: 16px;
      border-radius: 4px;
    }

    /* Mobile Responsive Rules */
    @media (max-width: 480px) {
      .stage {
        min-height: 230px;
        height: 235px;
      }
      .decompose-row {
        flex-direction: column;
        gap: 6px;
      }
      .badge-card {
        padding: 6px 10px;
        flex-direction: row;
        justify-content: space-between;
        align-items: center;
      }
      .card-formula {
        font-size: 0.95em;
      }
      .problem-strip {
        padding: 6px 8px;
      }
      .problem-math {
        font-size: 1.05em;
      }
      .nav-btn {
        font-size: 11px;
        padding: 6px 2px;
      }
      .balance-visual {
        gap: 5px;
        padding: 4px 6px;
      }
      .factor-box {
        padding: 2px 5px;
        font-size: 0.88em;
      }
      .neutral-pill {
        font-size: 10px;
        padding: 2px 6px;
      }
      .formula-box {
        font-size: 12px;
        padding: 6px 6px;
      }
    }
  </style>
</head>
<body>

<div class="diagram-container">
  
  <div class="problem-strip">
    <span class="problem-label">Problem</span>
    <span class="problem-math math">
      <span class="big-sym">∫</span> <var>x</var><sup>2</sup>(<var>x</var><sup>3</sup> + 5)<sup>1/2</sup> <var>dx</var>
    </span>
  </div>

  
  <div class="step-nav">
    <button class="nav-btn active" id="tab-0" type="button">
      <span class="num-badge">1</span> Identify
    </button>
    <button class="nav-btn" id="tab-1" type="button">
      <span class="num-badge">2</span> Balance
    </button>
    <button class="nav-btn" id="tab-2" type="button">
      <span class="num-badge">3</span> Integrate
    </button>
  </div>

  
  <div class="stage">
    
    <div class="step-panel active" id="panel-0">
      <div class="decompose-row">
        <div class="badge-card">
          <span class="badge-tag tag-blue">Inner Function</span>
          <span class="card-formula math"><var>g</var>(<var>x</var>) = <var>x</var><sup>3</sup> + 5</span>
        </div>
        <div class="badge-card">
          <span class="badge-tag tag-amber">Needed Derivative</span>
          <span class="card-formula math"><var>g</var>'(<var>x</var>) = 3<var>x</var><sup>2</sup></span>
        </div>
      </div>
      <div class="formula-box">
        <span>We have <strong class="math"><var>x</var><sup>2</sup> <var>dx</var></strong> ➔ missing factor of <strong style="color: #0284c7;">3</strong></span>
      </div>
    </div>

    
    <div class="step-panel" id="panel-1">
      <div class="balance-container">
        <div class="balance-visual">
          <div class="factor-box">
            <span class="math"><span class="frac"><span class="num">1</span><span class="den">3</span></span></span>
            <span class="factor-desc">(outside)</span>
          </div>
          <span class="multiply-sym">×</span>
          <div class="factor-box">
            <span class="math">3</span>
            <span class="factor-desc">(inside)</span>
          </div>
          <span class="multiply-sym">=</span>
          <span class="neutral-pill">Net Factor 1</span>
        </div>
      </div>
      <div class="formula-box">
        <span class="math">
          <span class="frac"><span class="num">1</span><span class="den">3</span></span> <span class="big-sym">∫</span> (3<var>x</var><sup>2</sup>)(<var>x</var><sup>3</sup> + 5)<sup>1/2</sup> <var>dx</var>
        </span>
      </div>
    </div>

    
    <div class="step-panel" id="panel-2">
      <div class="decompose-row">
        <div class="badge-card">
          <span class="badge-tag tag-green">Power Rule Integration</span>
          <span class="card-formula math">
            <span class="frac"><span class="num">1</span><span class="den">3</span></span> · 
            <span class="frac">
              <span class="num">(<var>x</var><sup>3</sup> + 5)<sup>3/2</sup></span>
              <span class="den"><span class="frac"><span class="num">3</span><span class="den">2</span></span></span>
            </span> + <var>C</var>
          </span>
        </div>
      </div>
      <div class="formula-box success-box">
        <span class="math" style="font-size: 1.08em; font-weight: 700;">
          = <span class="frac"><span class="num">2</span><span class="den">9</span></span>(<var>x</var><sup>3</sup> + 5)<sup>3/2</sup> + <var>C</var>
        </span>
      </div>
    </div>
  </div>

  
  <div class="controls">
    <button class="btn-ctrl" id="prev-btn" type="button">‹ Prev</button>
    <div class="dots-wrap">
      <div class="dot active" id="dot-0" title="Step 1"></div>
      <div class="dot" id="dot-1" title="Step 2"></div>
      <div class="dot" id="dot-2" title="Step 3"></div>
    </div>
    <button class="btn-ctrl" id="next-btn" type="button">Next ›</button>
  </div>
</div>

<script>
  (function() {
    const panels = [
      document.getElementById('panel-0'),
      document.getElementById('panel-1'),
      document.getElementById('panel-2')
    ];
    const tabs = [
      document.getElementById('tab-0'),
      document.getElementById('tab-1'),
      document.getElementById('tab-2')
    ];
    const dots = [
      document.getElementById('dot-0'),
      document.getElementById('dot-1'),
      document.getElementById('dot-2')
    ];

    let current = 0;

    function setStep(index) {
      current = (index + panels.length) % panels.length;
      panels.forEach((p, i) => p.classList.toggle('active', i === current));
      tabs.forEach((t, i) => t.classList.toggle('active', i === current));
      dots.forEach((d, i) => d.classList.toggle('active', i === current));
    }

    tabs.forEach((tab, i) => {
      tab.addEventListener('click', () => setStep(i));
    });

    dots.forEach((dot, i) => {
      dot.addEventListener('click', () => setStep(i));
    });

    document.getElementById('prev-btn').addEventListener('click', () => setStep(current - 1));
    document.getElementById('next-btn').addEventListener('click', () => setStep(current + 1));
  })();
</script>

</body>
</html>

Standard Algebraic Integrals and the Reverse Chain Rule

<!DOCTYPE html>
<html lang="en">
<head>
  <meta charset="UTF-8">
  <meta name="viewport" content="width=device-width, initial-scale=1.0">
  <style>
    * {
      box-sizing: border-box;
      margin: 0;
      padding: 0;
    }
    body {
      width: 100%;
      background: #f8f9fa;
      font-family: system-ui, -apple-system, BlinkMacSystemFont, "Segoe UI", Roboto, sans-serif;
      color: #1e2945;
      padding: 12px 8px;
      display: flex;
      flex-direction: column;
      align-items: center;
      justify-content: flex-start;
      min-height: 100%;
    }
    .main-layout {
      width: 100%;
      max-width: 480px;
      display: flex;
      flex-direction: column;
      gap: 12px;
    }
    .toggle-row {
      display: flex;
      background: #e2e8f0;
      padding: 3px;
      border-radius: 8px;
      gap: 4px;
      width: 100%;
    }
    .tab-btn {
      flex: 1;
      border: none;
      background: transparent;
      padding: 7px 10px;
      border-radius: 6px;
      font-size: 13px;
      font-weight: 700;
      color: #1e293b;
      cursor: pointer;
      transition: background 0.2s ease, color 0.2s ease, box-shadow 0.2s ease;
    }
    .tab-btn:hover:not(.active) {
      color: #0f172a;
      background: rgba(255, 255, 255, 0.5);
    }
    .tab-btn.active {
      background: #ffffff;
      color: #0f172a;
      box-shadow: 0 1px 3px rgba(0, 0, 0, 0.12), 0 1px 2px rgba(0, 0, 0, 0.08);
    }
    .display-section {
      display: flex;
      flex-direction: column;
      align-items: center;
      gap: 12px;
      width: 100%;
    }
    .equation-wrapper {
      width: 100%;
      display: flex;
      justify-content: center;
      align-items: center;
      background: #ffffff;
      border: 1px solid #e2e8f0;
      border-radius: 8px;
      padding: 8px 4px;
      overflow-x: auto;
      scrollbar-width: none;
    }
    .equation-wrapper::-webkit-scrollbar {
      display: none;
    }
    .equation-row {
      display: flex;
      align-items: center;
      justify-content: center;
      flex-wrap: nowrap;
      white-space: nowrap;
      gap: 2px;
      font-size: 14px;
      line-height: 1.2;
      text-align: center;
      user-select: none;
    }
    .math-term {
      display: inline-flex;
      align-items: center;
      white-space: nowrap;
    }
    .arrow-icon {
      color: #64748b;
      font-size: 14px;
      font-weight: bold;
      margin: 0 3px;
      flex-shrink: 0;
    }
    .math-sym {
      font-family: "Cambria Math", "Times New Roman", serif;
      font-style: italic;
      font-size: 17px;
      margin-right: 2px;
      flex-shrink: 0;
    }
    .plus-c {
      font-size: 13px;
      font-weight: 600;
      margin-left: 2px;
      flex-shrink: 0;
    }
    .badge {
      display: inline-flex;
      align-items: center;
      justify-content: center;
      padding: 2px 4px;
      border-radius: 4px;
      font-weight: 700;
      font-size: 12px;
      border: 1.5px solid transparent;
      transition: background-color 0.2s ease, border-color 0.2s ease;
      white-space: nowrap;
      flex-shrink: 0;
    }
    /* 3 Color Zones */
    .c-blue {
      background: #e0f2fe;
      color: #0369a1;
      border-color: #7dd3fc;
    }
    .c-orange {
      background: #ffedd5;
      color: #c2410c;
      border-color: #fdba74;
    }
    .c-green {
      background: #dcfce7;
      color: #15803d;
      border-color: #86efac;
    }

    /* Flow breakdown grid */
    .flow-grid {
      display: flex;
      flex-direction: column;
      gap: 8px;
      width: 100%;
    }
    .flow-item {
      display: flex;
      align-items: center;
      gap: 8px;
      background: #ffffff;
      border: 1px solid #e2e8f0;
      border-radius: 8px;
      padding: 8px 10px;
      font-size: 12.5px;
    }
    .flow-badge {
      min-width: 95px;
      text-align: center;
      font-size: 11.5px;
      padding: 2px 5px;
      flex-shrink: 0;
    }
    .flow-arrow {
      color: #94a3b8;
      font-size: 12px;
      font-weight: bold;
      flex-shrink: 0;
    }
    .flow-desc {
      font-weight: 600;
      flex: 1;
      font-size: 12px;
      line-height: 1.35;
    }

    @media (min-width: 400px) {
      body {
        padding: 14px 12px;
      }
      .tab-btn {
        font-size: 13.5px;
        padding: 8px 12px;
      }
      .equation-wrapper {
        padding: 10px 8px;
      }
      .equation-row {
        gap: 3px;
        font-size: 16px;
      }
      .math-sym {
        font-size: 20px;
      }
      .arrow-icon {
        font-size: 16px;
        margin: 0 4px;
      }
      .plus-c {
        font-size: 15px;
        margin-left: 4px;
      }
      .badge {
        padding: 3px 6px;
        font-size: 13.5px;
        border-radius: 5px;
      }
      .flow-item {
        padding: 9px 12px;
        gap: 10px;
      }
      .flow-badge {
        min-width: 108px;
        font-size: 12.5px;
        padding: 3px 6px;
      }
      .flow-desc {
        font-size: 13px;
      }
    }
  </style>
</head>
<body>
  <div class="main-layout">
    
    <div class="toggle-row">
      <button class="tab-btn active" id="tabGeneral">General Rule</button>
      <button class="tab-btn" id="tabExample">Example</button>
    </div>

    
    <div class="display-section">
      <div class="equation-wrapper">
        <div class="equation-row" id="eqContainer">
          
        </div>
      </div>

      
      <div class="flow-grid" id="guideGrid">
        
      </div>
    </div>
  </div>

  <script>
    const data = {
      general: {
        integrandHtml: `
          <span class="math-term"><span class="math-sym">&int;</span><span class="badge c-blue">f(</span><span class="badge c-orange">g(x)</span><span class="badge c-blue">)</span></span>
          <span class="math-term"><span class="badge c-green">g'(x) dx</span></span>
        `,
        resultHtml: `
          <span class="arrow-icon">&rarr;</span>
          <span class="math-term"><span class="badge c-blue">F(</span><span class="badge c-orange">g(x)</span><span class="badge c-blue">)</span></span>
          <span class="plus-c">+ C</span>
        `,
        rows: [
          {
            badgeClass: 'c-blue',
            badgeText: "f(...) &rarr; F(...)",
            desc: "Outer function: integrate to antiderivative"
          },
          {
            badgeClass: 'c-orange',
            badgeText: 'g(x)',
            desc: 'Inner function: stays intact inside'
          },
          {
            badgeClass: 'c-green',
            badgeText: "g'(x) dx",
            desc: 'Derivative factor: absorbed in integration'
          }
        ]
      },
      example: {
        integrandHtml: `
          <span class="math-term"><span class="math-sym">&int;</span><span class="badge c-orange">(x&sup2; + 1)</span><span class="badge c-blue">&sup3;</span></span>
          <span class="math-term"><span class="badge c-green">&middot; 2x dx</span></span>
        `,
        resultHtml: `
          <span class="arrow-icon">&rarr;</span>
          <span class="math-term"><span class="badge c-blue">&frac14;</span><span class="badge c-orange">(x&sup2; + 1)</span><span class="badge c-blue">&sup4;</span></span>
          <span class="plus-c">+ C</span>
        `,
        rows: [
          {
            badgeClass: 'c-blue',
            badgeText: "(&bull;)&sup3; &rarr; &frac14;(&bull;)&sup4;",
            desc: "Outer power rule: integrate &bull;&sup3; to &frac14;&bull;&sup4;"
          },
          {
            badgeClass: 'c-orange',
            badgeText: 'x&sup2; + 1',
            desc: 'Inner expression: unchanged inside'
          },
          {
            badgeClass: 'c-green',
            badgeText: '2x dx',
            desc: "Derivative of (x&sup2;+1): consumed into du"
          }
        ]
      }
    };

    const eqContainer = document.getElementById('eqContainer');
    const guideGrid = document.getElementById('guideGrid');
    const tabGeneral = document.getElementById('tabGeneral');
    const tabExample = document.getElementById('tabExample');

    function render(mode) {
      const item = data[mode];
      eqContainer.innerHTML = item.integrandHtml + item.resultHtml;
      
      guideGrid.innerHTML = item.rows.map(r => {
        const color = r.badgeClass === 'c-blue' ? '#0369a1' : (r.badgeClass === 'c-orange' ? '#c2410c' : '#15803d');
        return `
          <div class="flow-item">
            <span class="badge ${r.badgeClass} flow-badge">${r.badgeText}</span>
            <span class="flow-arrow">&rarr;</span>
            <span class="flow-desc" style="color: ${color}">${r.desc}</span>
          </div>
        `;
      }).join('');
    }

    tabGeneral.addEventListener('click', () => {
      tabGeneral.classList.add('active');
      tabExample.classList.remove('active');
      render('general');
    });

    tabExample.addEventListener('click', () => {
      tabExample.classList.add('active');
      tabGeneral.classList.remove('active');
      render('example');
    });

    // Initial render
    render('general');
  </script>
</body>
</html>

OGM.13.CAL.6

Aligned learning resources

Related diagrams