Unit 10 · Lesson 07 · Week 30 · Fundamental Theorem & Antiderivatives
Definite Integrals with 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 definite integrals with substitution through symbolic, graphical, numerical, and contextual evidence.
Learning intentions
- Define and use the central calculus language in definite integrals with 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
Definite Integrals with Substitution
Teacher Answer Key · Analyze, calculate, graph, or justify each result. Show reasoning, intervals, units, and technology evidence when used.
Name
Date
Evaluate ∫2x(x²+1)³ dx.
Answer: (x²+1)⁴/4+C
Reasoning: Let u=x²+1 and du=2x dx.
Reasoning: Let u=x²+1 and du=2x dx.
Evaluate ∫₀¹ 2x e^(x²) dx.
Answer: e−1
Reasoning: Use u=x² and change bounds from 0 to 1.
Reasoning: Use u=x² and change bounds from 0 to 1.
Choose u for ∫cos(3x)dx.
Answer: u=3x
Reasoning: The inner angle is the natural substitution.
Reasoning: The inner angle is the natural substitution.
Evaluate ∫cos(3x)dx.
Answer: (1/3)sin(3x)+C
Reasoning: Account for the inner derivative 3.
Reasoning: Account for the inner derivative 3.
Why change bounds in a definite substitution?
Answer: The integral is rewritten in the new variable.
Reasoning: Bounds must match the integration variable.
Reasoning: Bounds must match the integration variable.
How is substitution verified?
Answer: Differentiate the result.
Reasoning: The chain rule should recover the integrand.
Reasoning: The chain rule should recover the integrand.