6.8 Antiderivatives & Indefinite Integrals

Antiderivatives

F is an antiderivative of f if F'(x) = f(x). The indefinite integral ∫f(x) dx = F(x) + C — always add the constant, since the derivative of any constant is 0.

Key rules

Power rule for integrals: ∫x^n dx = x^(n+1)/(n+1) + C (n ≠ −1). Also ∫(1/x) dx = ln|x| + C, ∫cos x dx = sin x + C, ∫e^x dx = e^x + C.

Key terms

Antiderivative

F with F'(x) = f(x).

Indefinite integral

∫f dx = F(x) + C.

+ C

Needed because constants differentiate to 0.

Power rule (integrals)

∫x^n dx = x^(n+1)/(n+1) + C, n ≠ −1.

∫(1/x) dx

= ln|x| + C.

Try an AP-style question

∫ x⁴ dx = ?

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