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</head>
<body>

  <div class="header-bar">
    <div class="header-title">Constant Balancing in Integration</div>
  </div>

  
  <div class="integral-display" id="main-integral">
    <span>\int \color{#0d9488}{x^2} \sqrt{\color{#ea580c}{x^3 + 5}} \,\color{#0d9488}{dx}</span>
  </div>

  
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    <button class="step-btn active" id="btn0" onclick="setStep(0)">1. Split &amp; Identify</button>
    <button class="step-btn" id="btn1" onclick="setStep(1)">2. Balance Constant</button>
    <button class="step-btn" id="btn2" onclick="setStep(2)">3. Substitute <em>u</em></button>
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  <div class="breakdown-grid">
    <div class="card-box card-u">
      <span class="badge-label label-orange">Inside Power (<em>u</em>)</span>
      <div class="math-content" id="math-u">
        <div>u = x^3 + 5</div>
      </div>
    </div>
    
    <div class="card-box card-du">
      <span class="badge-label label-teal">Derivative Matching</span>
      <div class="math-content" id="math-du">
        <div>du = 3x^2 dx</div>
      </div>
    </div>
  </div>

  
  <div class="result-box">
    <span class="result-desc" id="sub-desc">Notice the remaining factor outside the square root is <em>x</em>² <em>dx</em>:</span>
    <div class="math-content" id="math-sub">
      \frac{1}{3}du = x^2 dx
    </div>
  </div>

  
  <div class="rules-grid">
    <div class="rule-card rule-bad">
      <div class="rule-title">✕ Never Pull Variables Out</div>
      <div class="rule-math" id="math-bad">\int x^2 \dots dx \neq x^2 \int \dots dx</div>
      <div class="rule-note">Variables must stay inside the integral</div>
    </div>

    <div class="rule-card rule-good">
      <div class="rule-title">✓ Constants Move Freely</div>
      <div class="rule-math" id="math-good">\int \frac{1}{3} \dots du = \frac{1}{3} \int \dots du</div>
      <div class="rule-note">Factor out constant multiplier <sup>1</sup>/<sub>3</sub></div>
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</html>

Integration by Substitution for Indefinite Integrals

<!DOCTYPE html>
<html lang="en">
<head>
  <meta charset="UTF-8">
  <meta name="viewport" content="width=device-width, initial-scale=1.0">
  <title>Chain Rule vs u-Substitution</title>
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</head>
<body>
  <div class="diagram-wrap">
    
    <div class="nav-bar">
      <button class="nav-btn active" id="btn-0" onclick="setStep(0)">Overview</button>
      <button class="nav-btn" id="btn-1" onclick="setStep(1)">1. Chain Rule</button>
      <button class="nav-btn" id="btn-2" onclick="setStep(2)">2. <em>u</em>-Bridge</button>
      <button class="nav-btn" id="btn-3" onclick="setStep(3)">3. <em>u</em>-Sub</button>
    </div>

    
    <div class="flow-grid">
      
      <div class="math-node node-top" id="node-top">
        <span class="badge badge-top" id="lbl-top">Composite Function</span>
        <div class="math math-exp-main" id="exp-top">
          <em>F</em>(<em>g</em>(<em>x</em>))
        </div>
      </div>

      
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        <span class="badge badge-diff">
          <span class="dir-arrow">↓</span> Differentiate
        </span>
        <div class="math math-diff-exp">
          <span class="math-frac">
            <span class="num"><em>d</em></span>
            <span class="denom"><em>dx</em></span>
          </span>
          <span>[<em>F</em>(<em>g</em>(<em>x</em>))]</span>
        </div>
        <span style="font-size: 10.5px; color: #b45309; font-weight: 600;">via Chain Rule</span>
      </div>

      
      <div class="math-node node-sub" id="node-sub">
        <span class="badge badge-sub">
          <span class="dir-arrow">↑</span> Integrate via <span class="nowrap"><em>u</em>-Substitution</span>
        </span>
        <div class="math math-sub-exp">
          <div class="bridge-row">
            <span>Let <em>u</em> = <em>g</em>(<em>x</em>) <span class="op">⇒</span> <em>du</em> = <em>g</em>′(<em>x</em>) <em>dx</em></span>
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          <div class="bridge-row">
            <span>∫ <em>f</em>(<em>u</em>) <em>du</em> = <em>F</em>(<em>u</em>) + <em>C</em></span>
          </div>
        </div>
      </div>

      
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        <span class="badge badge-bottom" id="lbl-bottom">Derivative / Integrand</span>
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          <em>f</em>(<em>g</em>(<em>x</em>)) · <em>g</em>′(<em>x</em>)
        </div>
      </div>
    </div>

    
    <div class="action-banner" id="status-banner">
      Differentiation via Chain Rule and Integration via <span class="nowrap"><em>u</em>-Substitution</span> are inverse operations.
    </div>
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<body>
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      <div class="legend-item"><span class="legend-dot bg-outer"></span> Outer: <span class="c-outer">F, f</span></div>
      <div class="legend-item"><span class="legend-dot bg-inner"></span> Inner: <span class="c-inner">g(x) = u</span></div>
      <div class="legend-item"><span class="legend-dot bg-deriv"></span> Derivative: <span class="c-deriv">g'(x)dx = du</span></div>
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    <div class="comparison-grid">
      
      
      <div class="panel">
        <div class="panel-header">
          <span class="panel-title">Forward: Chain Rule</span>
          <span class="tag tag-fwd">d/dx</span>
        </div>

        <div class="math-block">
          <span class="math-op">d/dx [</span>
          <span class="pill-outer">F</span>
          <span class="math-op">(</span>
          <span class="pill-inner">g(x)</span>
          <span class="math-op">) ]</span>
          <span class="math-op">=</span>
          <span class="pill-outer">f</span>
          <span class="math-op">(</span>
          <span class="pill-inner">g(x)</span>
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          <span class="math-op">·</span>
          <span class="pill-deriv">g'(x)</span>
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            <span class="c-outer">Outer</span>: <span class="math-sym">F'(...) → f(...)</span>
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            <span class="c-deriv">Multiply</span>: <span class="math-sym">· g'(x)</span>
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        </div>
      </div>

      
      <div class="panel">
        <div class="panel-header">
          <span class="panel-title">Reverse: u-Substitution</span>
          <span class="tag tag-rev">∫ dx</span>
        </div>

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          <div class="step-row active" id="step0" onclick="goToStep(0)">
            <span class="step-num">1</span>
            <div class="step-expr">
              <span class="integral-symbol">∫</span>
              <span class="c-outer">f</span>(<span class="pill-inner">g(x)</span>) · <span class="pill-deriv">g'(x) dx</span>
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            <span class="arrow-icon">→</span>
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            <span class="step-num">2</span>
            <div class="step-expr">
              <span class="integral-symbol">∫</span>
              <span class="c-outer">f</span>(<span class="pill-inner">u</span>) <span class="pill-deriv">du</span>
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            <span class="arrow-icon">→</span>
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            <span class="step-num">3</span>
            <div class="step-expr">
              <span class="pill-outer">F</span>(<span class="pill-inner">u</span>) + C = <span class="pill-outer">F</span>(<span class="pill-inner">g(x)</span>) + C
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            <span class="arrow-icon">✓</span>
          </div>
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          <div class="map-cell">
            <span class="pill-inner">u = g(x)</span>
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  <div class="header-bar">
    <div class="title">u-Substitution: ∫ 2x cos(x²) dx</div>
    <div id="stepPill" class="step-pill">Step 1 of 5</div>
  </div>

  <div class="nav-tabs" id="navTabs">
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    <button class="tab-btn" data-step="1">2. Define</button>
    <button class="tab-btn" data-step="2">3. Substitute</button>
    <button class="tab-btn" data-step="3">4. Integrate</button>
    <button class="tab-btn" data-step="4">5. Back-Sub</button>
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  <div class="stage-container">
    
    <div class="step-card active" id="card-0">
      <div class="label-tag">Step 1: Identify Parts</div>
      <div class="math-display" id="math-1"></div>
      <div class="sub-note">
        Inner function is <span class="hl-amber">x²</span> and derivative factor is <span class="hl-green">2x dx</span>
      </div>
    </div>

    
    <div class="step-card" id="card-1">
      <div class="label-tag">Step 2: Substitution Dictionary</div>
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          <span>u = x²</span>
        </div>
        <div class="dict-box du" id="math-du">
          <span>du = 2x dx</span>
        </div>
      </div>
      <div class="sub-note">
        Let <span class="hl-amber">u = x²</span>, then differentiate: <span class="hl-green">du = 2x dx</span>
      </div>
    </div>

    
    <div class="step-card" id="card-2">
      <div class="label-tag">Step 3: Transformed Integral</div>
      <div class="math-display" id="math-3"></div>
      <div class="sub-note">
        Substitute <span class="hl-amber">u</span> for <span class="hl-amber">x²</span> and <span class="hl-green">du</span> for <span class="hl-green">2x dx</span>
      </div>
    </div>

    
    <div class="step-card" id="card-3">
      <div class="label-tag">Step 4: Integrate in terms of u</div>
      <div class="math-display" id="math-4"></div>
      <div class="sub-note">
        Standard antiderivative rule: <span class="hl-amber">∫ cos(u) du = sin(u) + C</span>
      </div>
    </div>

    
    <div class="step-card" id="card-4">
      <div class="label-tag">Step 5: Back-Substitute</div>
      <div class="math-display" id="math-5"></div>
      <div class="success-badge">
        <span>✓</span> Back-substitute u = x²
      </div>
    </div>
  </div>

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          katex.render(tex, el, { displayMode: false, throwOnError: false });
          return;
        } catch (e) {
          console.warn("KaTeX render error:", e);
        }
      }
      if (fallbackHtml) {
        el.innerHTML = fallbackHtml;
      }
    }

    function renderAllMath() {
      renderMathSafely(stepsData[0].id, stepsData[0].tex, stepsData[0].fallback);
      renderMathSafely(stepsData[1].uId, stepsData[1].uTex, stepsData[1].uFallback);
      renderMathSafely(stepsData[1].duId, stepsData[1].duTex, stepsData[1].duFallback);
      renderMathSafely(stepsData[2].id, stepsData[2].tex, stepsData[2].fallback);
      renderMathSafely(stepsData[3].id, stepsData[3].tex, stepsData[3].fallback);
      renderMathSafely(stepsData[4].id, stepsData[4].tex, stepsData[4].fallback);
    }

    renderAllMath();

    let currentStep = 0;
    const totalSteps = 5;
    let isPlaying = true;
    let timer = null;

    const cards = Array.from(document.querySelectorAll('.step-card'));
    const tabs = Array.from(document.querySelectorAll('.tab-btn'));
    const stepPill = document.getElementById('stepPill');
    const playBtn = document.getElementById('playBtn');
    const prevBtn = document.getElementById('prevBtn');
    const nextBtn = document.getElementById('nextBtn');

    function updateUI() {
      cards.forEach((card, idx) => {
        card.classList.toggle('active', idx === currentStep);
      });
      tabs.forEach((tab, idx) => {
        tab.classList.toggle('active', idx === currentStep);
      });
      stepPill.textContent = `Step ${currentStep + 1} of ${totalSteps}`;
    }

    function goToStep(step) {
      currentStep = (step + totalSteps) % totalSteps;
      updateUI();
    }

    function startAutoPlay() {
      stopAutoPlay();
      isPlaying = true;
      playBtn.textContent = '⏸';
      timer = setInterval(() => {
        goToStep(currentStep + 1);
      }, 3500);
    }

    function stopAutoPlay() {
      isPlaying = false;
      playBtn.textContent = '▶';
      if (timer) clearInterval(timer);
    }

    function toggleAutoPlay() {
      if (isPlaying) {
        stopAutoPlay();
      } else {
        startAutoPlay();
      }
    }

    tabs.forEach((tab) => {
      tab.addEventListener('click', () => {
        stopAutoPlay();
        goToStep(parseInt(tab.dataset.step, 10));
      });
    });

    prevBtn.addEventListener('click', () => {
      stopAutoPlay();
      goToStep(currentStep - 1);
    });

    nextBtn.addEventListener('click', () => {
      stopAutoPlay();
      goToStep(currentStep + 1);
    });

    playBtn.addEventListener('click', toggleAutoPlay);

    startAutoPlay();
  </script>
</body>
</html>

OGM.12.CAL.5

Aligned learning resources

Related diagrams