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C24 · Unit C6 Integration and accumulation

Antiderivatives and definite evaluation

CoreReview editionNot yet human-reviewed
Time
About 45 minutes, untimed
Course framework reference
AP 6.7–6.8
Builds on
Part of

Progress on this lesson counts in every pathway listed.

Understand the idea

An antiderivative F satisfies F minutes equals f. The family includes an arbitrary constant because differentiation loses constants. The evaluation form of FTC states ∫ₐᵇf(x)dx equals F(b) minus F(a) for continuous f. The power antiderivative is x^(n+1)/(n+1) for n is not equal to minus 1; the exceptional reciprocal integrates to ln|x| on intervals excluding zero.

Worked example

∫ sub 1 cubed (2x minus 1)dx equals [x squared minus x] sub 1 cubed equals 6 minus 0 equals 6. An indefinite integral would be x squared minus x+C; the constant cancels in a definite evaluation. Always differentiate a proposed antiderivative as a quick check.

Explore the mathematics

Antiderivatives and definite evaluation

Reverse differentiation and distinguish indefinite from definite integration.

Setting 3 of 5

Scroll sideways to see the full diagram.

What function has derivative x²?xy123Marked input 1.5; displayed curve height 2.25Area beneath y=x² on [0,2]

Predict before continuing

What function has derivative x²?

Read the complete explanation
  1. What function has derivative x²?
  2. Raise the exponent and divide by the new exponent: F(x)=x³/3+C.
  3. For a definite integral, subtract F(0) from F(2).
  4. The result is 8/3. The constant cancels; the family of indefinite antiderivatives still needs +C.
See the idea: Antiderivatives and definite evaluation
See the idea: Antiderivatives and definite evaluationBetween x=1 and x=3, the trapezoid under 2x−1 has parallel heights 1 and 5 and width 2. Its area is 6, agreeing with [x²−x]₁³. Only that interval is integrated.xy0012243648

Between x=1 and x=3, the trapezoid under 2x−1 has parallel heights 1 and 5 and width 2. Its area is 6, agreeing with [x²−x]₁³. Only that interval is integrated.

Discuss: explain what each label represents, then draw a related example with different values.

A common misconception

Do not use the power formula at n equals minus 1. A definite integral has a value; it does not retain +C.

Investigate and explain

Use a derivative to verify each of three antiderivatives, including one with a logarithm. State each domain.
Identify one step in the worked example, name the mathematical fact that permits it, and state its domain conditions.