Section 5.1: Antiderivatives and Indefinite Integrals

Antiderivatives

A function F is an antiderivative of a function f if F'(x) = f(x).

Equal Derivatives

x = 0

k = 1

Indefinite Integrals

We use the symbol

f(x) dx

called the indefinite integral, to represent the family of all antiderivatives of f(x), and we write:

f(x) dx = F(x) + CifF'(x) = f(x)

Indefinite Integrals of Basic Functions

For C a constant:

  1. xn dx = xn+1/(n + 1) + C, n ≠ -1
  2. ex dx = ex + C
  3. 1/x dx = ln |x| + C, x ≠ 0

Properties of Indefinite Integrals

For k a constant:

  1. k f(x) dx = k f(x) dx
  2. [f(x) ± g(x)] dx = f(x) dx ± g(x) dx