Antiderivatives & the Definite Integral (FTC)

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Skill: integrals

Antiderivatives & the Definite Integral (FTC)

Undo the derivative: antiderivatives rebuild a function from its rate of change. The Fundamental Theorem of Calculus turns area problems into antiderivative evaluations — with net area keeping honest count below the axis.

1 Antiderivatives: undoing the derivative

An antiderivative of f is any function F with F′ = f.
Because constants differentiate to zero, antiderivatives come in whole families — that’s what the
+ C is for.

Power rule for antiderivatives
∫ xn dx = xn+1/(n+1) + C   (n ≠ −1)

Raise the power by one, divide by the new power, add C. Then check yourself: differentiate the answer and you must get the integrand back.

Example ∫(6x2 + 4x) dx
  1. ∫ 6x2 dx = 6 · x3/3 = 2x3.
  2. ∫ 4x dx = 4 · x2/2 = 2x2.
  3. Answer: 2x3 + 2x2 + C. Check: d/dx gives 6x2 + 4x. Correct.
Example ∫ 5cos(2x) dx

Work backwards from d/dx[sin(2x)] = 2cos(2x): we need an extra factor. ∫ 5cos(2x) dx = (5/2)sin(2x) + C.
Check: d/dx[(5/2)sin(2x)] = (5/2)(2cos(2x)) = 5cos(2x).

2 The Fundamental Theorem of Calculus

Differentiation and integration are inverses. The FTC turns a definite integral — a limit of
Riemann sums — into a simple subtraction.

FTC (evaluation part)
∫ab f(x) dx = F(b) − F(a)

If F is any antiderivative of f, the definite integral is just F at the top minus F at the bottom. No limits, no sums.

Example ∫13 (4x + 1) dx
  1. Antiderivative: F(x) = 2x2 + x.
  2. Evaluate: F(3) − F(1) = (18 + 3) − (2 + 1) = 21 − 3 = 18.
y = x² Riemann rectangles approach the true area as n grows

Riemann sums trap the area under the curve; the FTC computes it exactly via antiderivatives.

3 Net area: below the axis counts negative

A definite integral measures net (signed) area: regions above the x-axis add,
regions below subtract. If you want total geometric area, integrate the absolute value — or split at the zeros.

Example ∫−23 x dx
  1. Antiderivative: x2/2.
  2. (9/2) − (4/2) = 5/2.

Geometrically: a triangle of area 2 below the axis (negative) and a triangle of area 9/2 above:
9/2 − 2 = 5/2. The total area would be 9/2 + 2 = 13/2 — a different question.

4 Common mistakes

Three traps, each with the wrong version and the fix.

1. Forgetting the + C

Wrong: ∫ 2x dx = x2.   
Right: ∫ 2x dx = x2 + C. Every antiderivative needs the constant family — indefinite integrals always end with + C.

2. Treating below-axis area as positive

Wrong: “∫−11 x3 dx is the total area, so it’s positive.”   
Right: it’s net area: the odd symmetry cancels to exactly 0. Total area would be 1/2.

3. Raising the power but forgetting to divide

Wrong: ∫ x3 dx = x4 + C.   
Right: ∫ x3 dx = x4/4 + C. Divide by the new power — differentiate back to check.

Key vocabulary

  • Antiderivative — F with F′ = f; the indefinite integral ∫ f(x) dx.
  • + C — the constant of integration; names the whole family of antiderivatives.
  • Definite integral — ∫ab f(x) dx, a number (net area).
  • FTC — ∫ab f(x) dx = F(b) − F(a).
  • Net area — signed area: above the axis positive, below negative.
  • Riemann sum — area approximated by rectangles; the integral is its limit.

5 Quick checks

Cover the answers, decide, then reveal.

1. What is ∫ 3x2 dx?

x3 + C. Raise to x3, divide by 3: 3 · x3/3 = x3. Don’t forget + C.
2. ∫02 3x2 dx = ?

8. Antiderivative x3; F(2) − F(0) = 8 − 0 = 8.
3. True or false: ∫−22 x dx = 0 means there is no area between the curve and the axis.

False. The integral is net area: two triangles of area 2 cancel. The total geometric area is 4.
Practice now
Try the FTC evaluator
Open the Riemann lab
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