1. 5.a – Antiderivatives and The Indefinite Integral

2. Antiderivatives Definition A function F is called an antiderivative of f if F ′(x) = f (x) for all x on an interval I. Theorem If F is an antiderivative of f on an interval I, then the most general antiderivative of f on I is the family of functions given by F(x) + c, where c is an arbitrary constant. F(x) + c is a called a family of antiderivatives. If c is known, the antiderivative is called a specific antiderivative.

3. The Indefinite Integral Definition The general antiderivative, F(x), of a function f (x) can be represented by an indefinite integral Like the derivative, the dx denotes the variable with which we are anti-differentiating.

4. Examples Evaluate the indefinite integral (that is, determine the general anti-derivative) of the flowing functions. Use number three to develop a formula for

5. Properties of the Indefinite Integral Let c be a constant. Note: Always simplify the integrand before evaluating an integral.

6. Examples Evaluate the indefinite integral (that is, determine the general anti-derivative) of the flowing functions.

7. Examples Determine f if …

8. Example 8.A particle is moving according to the function a(t) = cos t + sin t [ft/sec 2 ] where s(0) = 0 and v(0) = 5. Find the position function of this particle.

9. Example 9.The graph of a derivative of some function is given below. Sketch a possible graph of the function. (a) (b)

10. Table of Basic Indefinite Integrals

12. Examples Evaluate the indefinite integrals.