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

• If the derivatives of two functions are equal on an open interval (a, b), then the functions differ by at most a constant.
• Symbolically, if F and G are differentiable functions on the interval (a, b) and F'(x) = G'(x) for all x in (a, b), then F(x) = G(x) + k for some constant k.

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)

• The symbol ∫ is called an integral sign.
• The function f(x) is called the integrand.
• The arbitrary constant C is called the constant of integration.

Indefinite Integrals of Basic Functions

For C a constant:

1. ∫ xⁿ dx = xⁿ⁺¹/(n + 1) + C, n ≠ -1
2. ∫ e^x dx = e^x + 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