Indefinite Integrals
An indefinite integral represents the family of all antiderivatives of a function. The result is a function plus an arbitrary constant C.
Definition
Indefinite Integral Notation
$$\int f(x) \, dx = F(x) + C$$
Where F'(x) = f(x) and C is the constant of integration
Understanding Antiderivatives
An antiderivative of f(x) is any function F(x) whose derivative equals f(x). The "+ C" accounts for all possible constants lost during differentiation.
Finding C with Initial Conditions
Example
Find f(x) if f'(x) = 3x² and f(1) = 5
- General antiderivative: f(x) = x³ + C
- Use f(1) = 5: 1 + C = 5, so C = 4
- Answer: f(x) = x³ + 4
Verifying Your Answer
Always differentiate your result to verify—you should get back the original integrand.