Calculus guide
Integration Basics: Antiderivatives and Definite Integrals
Understand antiderivatives, constants of integration, and how definite integrals measure signed accumulation.
9 min readBeginnerUpdated 2026-07-12
What you will learn
- Reverse the power rule
- Include the constant of integration
- Evaluate a definite integral with the Fundamental Theorem
Concept 1
An indefinite integral is a family of functions
An antiderivative F satisfies F'(x)=f(x). Because every constant disappears when differentiated, an indefinite integral must include +C.
Concept 2
Reverse the power rule
Increase the exponent by one and divide by the new exponent. The special case n=−1 produces a logarithm rather than the power formula.
Concept 3
Definite integrals use endpoint values
Find an antiderivative and subtract its value at the lower endpoint from its value at the upper endpoint. The result is signed area: regions below the axis contribute negatively.
Worked example
Evaluate ∫₀¹ x² dx
- 1Use the power rule to find x³/3.
- 2Evaluate at x=1 and x=0.
- 3Subtract the lower value from the upper value.
Answer
Common mistakes
- Leaving out +C for an indefinite integral
- Using the power rule when n=−1
- Adding endpoint values instead of subtracting F(a) from F(b)
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