6.8 Finding Antiderivatives and Indefinite Integrals: Basic Rules and Notation

Antiderivatives and indefinite integrals

An antiderivative of f(x) is a function F(x) such that F'(x) = f(x). Since constants vanish when differentiated, every antiderivative comes in a family: ∫f(x)dx = F(x) + C, where C is the constant of integration. Indefinite integral notation (∫...dx) represents this entire family of functions, not a single value like a definite integral.

Basic integration rules

The power rule for integration: ∫xⁿdx = xⁿ⁺¹/(n+1) + C (n ≠ -1). ∫kdx = kx + C. The sum/difference rule and constant multiple rule let you integrate term-by-term: ∫[f(x)±g(x)]dx = ∫f(x)dx ± ∫g(x)dx. These mirror differentiation rules in reverse.

Common function antiderivatives

Memorize: ∫eˣdx = eˣ+C, ∫(1/x)dx = ln|x|+C, ∫sin(x)dx = -cos(x)+C, ∫cos(x)dx = sin(x)+C, ∫sec²(x)dx = tan(x)+C. Always check your answer by differentiating the result — it should match the original integrand exactly.

Key terms

Antiderivative

A function F(x) such that F'(x) = f(x); the reverse process of differentiation.

Power rule for integration

∫xⁿdx = xⁿ⁺¹/(n+1) + C, valid for n ≠ -1.

Constant of integration (C)

Added to every indefinite integral because constants vanish under differentiation, representing a family of solutions.

∫(1/x)dx

ln|x| + C — the special case where the power rule doesn't apply (n = -1).

How to check an antiderivative

Differentiate your result — it should exactly equal the original integrand.

Try an AP-style question

∫ x⁴ dx = ?

1. 5x⁵ + C
2. x⁵/5 + C
3. x⁵ + C
4. 4x³ + C