Section 6.1—Antiderivatives

Problem: Given a function f(x) = 2x, find another function, F(x), such that F’(x) = f(x).

I was thinking of the function F(x) = x² + 5e

Definition: If F’(x) = f(x), then F(x) is called an antiderivative of f(x).

Theorem: If F’(x) = G’(x), then F(x) = G(x) + C.

Definition: The collection of all antiderivatives of a function f(x) is the indefinite integral of f(x) and is denoted by

If we know an antiderivative of f(x) is F(x), then we write

Rules of Integration:

Power Rule

Because

Constant times a function rule:

Sum or difference rule:

Exponential function rule:

Rule for x ^-1

Examples: Find the antiderivatives:

Example: Find the revenue function for a toy manufacturer if the marginal revenue, in dollars per plate, is given by

MR(x) = 200 – 0.26x, where x is the number of plates sold.

I know that, if R(x) is the revenue function, then R’(x) is the marginal revenue.

Then

If I sell 0 plates, what is my revenue? 0 dollars

So, R(0) = 0

R(0) = 200(0) – 0.13(0)² + C = 0; then C = 0

So, the revenue function is R(x) = 200x – 0.13x² dollars when x plates are sold