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 + e

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

I know: If R(x) gives the revenue (in dollars), then R ‘(x) = MR(x).

That means that the antiderivative of marginal revenue is the revenue function.

We can see that, if we don’t sell any plates, our revenue would be 0; thus R(0) = 0

200(0) – 0.13(0)² + C = 0 or C = 0

So our revenue function would be R(x) = 200x – 0.13x² dollars when x plates are sold