Unit 10 · Lesson 06 · Week 30 · Fundamental Theorem & Antiderivatives
Basic u-Substitution
Substitution reverses the chain rule. A successful substitution matches an inner expression with its derivative factor and transforms limits when working with a definite integral. This lesson emphasizes basic u-substitution through symbolic, graphical, numerical, and contextual evidence.
Learning intentions
- Define and use the central calculus language in basic u-substitution.
- Move among symbolic, graphical, numerical, and contextual representations.
- Show complete reasoning with correct notation, domain conditions, units, and intervals.
- Verify a conclusion using a second method or representation.
Essential question
How does the Fundamental Theorem connect instantaneous rate with accumulated change?
Visual model
Substitution reverses the chain rule. A successful substitution matches an inner expression with its derivative factor and transforms limits when working with a definite integral.
u=g(x)du=g'(x)dx
Worked reasoning
Misconception check: A definite integral cannot be evaluated with x-bounds after the integrand has been changed to u.
Three learning pathways
Foundation
Reconstruct definitions, annotate a worked example, and solve representative problems with guided prompts.
Core
Complete mixed representations, justify every major step, and verify exact and approximate results.
Extended
Generalize a result, analyze an error or edge case, and connect the lesson to modelling or proof-informed reasoning.
Communication and modelling
State what is known, identify the theorem or structure used, show symbolic work, interpret the result in context, and explain how the conclusion was checked. Calculator or CAD evidence must include settings, units, and a written interpretation.
Khaemenes Academy · Open-Age Calculus I
Basic u-Substitution
Student Worksheet · Analyze, calculate, graph, or justify each result. Show reasoning, intervals, units, and technology evidence when used.
Name
Date
Evaluate ∫2x(x²+1)³ dx.
Evaluate ∫₀¹ 2x e^(x²) dx.
Choose u for ∫cos(3x)dx.
Evaluate ∫cos(3x)dx.
Why change bounds in a definite substitution?
How is substitution verified?
Lesson record
Mark this lesson complete only after the worksheet, correction, and reflection are finished.