Antiderivatives & the Fundamental Theorem

You’ll be able to

Find antiderivatives using the reverse power rule and basic formulas
Evaluate definite integrals with the Fundamental Theorem of Calculus
Include the constant of integration for indefinite integrals

The antiderivative reverses differentiation

An antiderivative F of f is a function whose derivative is f: F′ = f. Finding antiderivatives is differentiation run backward. The reverse power rule is the main tool: ∫xⁿ dx = xⁿ⁺¹/(n + 1) + C for n ≠ −1 — raise the exponent by one and divide by the new exponent. Because any constant differentiates to zero, an indefinite integral always carries a + C, the constant of integration.

The Fundamental Theorem (evaluation part)

The Fundamental Theorem of Calculus ties the two big ideas of the course together. Its evaluation part: if F is any antiderivative of f, then ∫[a to b] f(x) dx = F(b) − F(a). To evaluate a definite integral, find an antiderivative, plug in the top and bottom limits, and subtract. The constant C cancels in the subtraction, so it is omitted for definite integrals.

Reverse power rule

∫xⁿ dx = xⁿ⁺¹/(n + 1) + C (n ≠ −1)

For n = −1, ∫(1/x) dx = ln|x| + C. Also ∫eˣ dx = eˣ + C and ∫cos x dx = sin x + C.

Fundamental Theorem of Calculus (Part 2)

∫[a to b] f(x) dx = F(b) − F(a), where F′ = f

Find an antiderivative, evaluate at the upper limit minus the lower limit. No + C needed for a definite integral.

Worked example

Evaluate ∫[1 to 3] (2x + 1) dx.

1.Find an antiderivative: ∫(2x + 1) dx = x² + x (+ C, dropped for a definite integral).
2.Evaluate at the upper limit x = 3: 3² + 3 = 9 + 3 = 12.
3.Evaluate at the lower limit x = 1: 1² + 1 = 2.
4.Subtract: F(3) − F(1) = 12 − 2.

Answer: ∫[1 to 3] (2x + 1) dx = 10.

Watch out

Never forget the + C on an indefinite integral — it represents an entire family of antiderivatives. For a definite integral you correctly drop it, because it cancels in F(b) − F(a). Match the +C to the kind of integral.

Checkpoint

What is ∫x³ dx?

Checkpoint

Evaluate ∫[0 to 2] 3x² dx.

On the exam

Verify any antiderivative by differentiating it back — if F′ returns the integrand, F is correct. This quick check catches reverse-power-rule slips (like forgetting to divide by the new exponent) before they cost points.

Answer the 2 checkpoints as you read.