Integrals

Antiderivatives

Finding antiderivatives using basic integration rules.

Advanced20 min lesson2 min readUpdated August 12, 2026Author not yet attributed

Prerequisites

The Definite Integral

Undoing Differentiation

Definition — Antiderivative

An antiderivative of f is a function F with F' = f. Reversing the power rule: ∫xⁿdx = xⁿ⁺¹/(n+1) + C (for n ≠ −1). The constant C accounts for every antiderivative differing only by a constant, since a constant's derivative is always 0.

Worked Example — Finding an Antiderivative

Find ∫(4x³ − 6x + 5)dx. Reversing the power rule term by term: x⁴ − 3x² + 5x + C. Check by differentiating: d/dx[x⁴−3x²+5x+C] = 4x³ − 6x + 5 — matching the original integrand.

Worked Example — Antiderivatives of Radicals and Fractions

Find ∫(3√x + 2/x²)dx. Rewrite as 3x^(1/2) + 2x^(−2). Antiderivative: 3 · x^(3/2)/(3/2) + 2 · x^(−1)/(−1) = 2x^(3/2) − 2/x + C. Check: d/dx[2x^(3/2)] = 3x^(1/2) = 3√x, and d/dx[−2/x] = 2/x² — both match.

Tip

Always verify an antiderivative by differentiating your result — it should exactly reproduce the original integrand, term for term.

Common Mistakes

Forgetting the +C on an indefinite integral.
Since any constant's derivative is 0, infinitely many functions share the same derivative — +C represents that whole family, and dropping it loses that fact.
Antidifferentiating x^(−2) as x^(−1)/(−1) without adjusting the sign correctly.
x^(−2) integrates to x^(−1)/(−1) = −x^(−1) = −1/x — dropping or mishandling the negative sign flips the result.

Key Takeaways

An antiderivative reverses differentiation; the power rule reverses to ∫xⁿdx = xⁿ⁺¹/(n+1) + C.
Every indefinite integral includes +C, representing the whole family of functions sharing that derivative.

Summary

The next lesson connects antiderivatives directly to definite integrals through the Fundamental Theorem of Calculus.

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