Unit 10 · Lesson 04 · Week 29 · Fundamental Theorem & Antiderivatives
Antiderivatives and Indefinite Integrals
An antiderivative reverses differentiation. Indefinite integrals represent families of functions and therefore require a constant of integration; initial conditions select one member. This lesson emphasizes antiderivatives and indefinite integrals through symbolic, graphical, numerical, and contextual evidence.
Learning intentions
- Define and use the central calculus language in antiderivatives and indefinite integrals.
- 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
An antiderivative reverses differentiation. Indefinite integrals represent families of functions and therefore require a constant of integration; initial conditions select one member.
\int x^n dx=\frac{x^{n+1}}{n+1}+CF'(x)=f(x)
Worked reasoning
Misconception check: The constant of integration is required for indefinite integrals because many functions share the same derivative.
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
Antiderivatives and Indefinite Integrals
Teacher Answer Key · Analyze, calculate, graph, or justify each result. Show reasoning, intervals, units, and technology evidence when used.
Name
Date
Find ∫x⁴ dx.
Answer: x⁵/5+C
Reasoning: Reverse the power rule.
Reasoning: Reverse the power rule.
Find ∫(3x²−2)dx.
Answer: x³−2x+C
Reasoning: Integrate term by term.
Reasoning: Integrate term by term.
Find F if F'=2x and F(1)=5.
Answer: F(x)=x²+4
Reasoning: The general form x²+C gives C=4.
Reasoning: The general form x²+C gives C=4.
Why include +C?
Answer: All constants have derivative zero.
Reasoning: Indefinite integration produces a family.
Reasoning: Indefinite integration produces a family.
Find ∫e^x dx.
Answer: e^x+C
Reasoning: e^x is its own derivative.
Reasoning: e^x is its own derivative.
Check an antiderivative.
Answer: Differentiate it.
Reasoning: The derivative should reproduce the integrand.
Reasoning: The derivative should reproduce the integrand.