ANTIDERIVATIVES AND INDEFINITE INTEGRATION

The opposite of a derivative is called an antiderivative or integral.

Definition: A function F is an antiderivative of f on an interval I if for all x in I. EX #1: Antiderivatives differ only by a constant, C: C is called the constant of integration

Family of all antiderivatives of f(x) = 2x and the general solution of the differential equation

A differential equation in x and y is an equation that involves x, y, and derivatives of y. For example:

and

1 EX #2: Solving a Differential Equation

EX #3: Notation for antiderivatives:

The operation of finding all solutions of this equation is called antidifferentiation or indefinite integration denoted by sign.

differential form

General Solution is denoted by:

Variable of Constant of Integration Integration

Integrand

read as the antiderivative of f with respect to x. So, the differential dx serves to identify x as the variable of integration. The term indefinite integral is a synonym for antiderivative.

2 BASIC INTEGRATION RULES Integration is the “inverse” of differentiation.

Differentiation is the “inverse” of integration.

Differentiation Formula Integration Formula

POWER RULE:

3 EX #4: Applying Basic Rules

EX #5: Rewriting Before Integrating

Original Integral Rewrite Integrate Simplify

4 EX # 6: Polynomial Functions

A.

B.

C.

EX #7: Integrate By Rewriting

5 EX #8: Solve differential equations subject to given conditions. Given: and

6 EX #9: A ball is thrown upward with an initial velocity of 64 feet per second from an initial height of 80 feet.

A. Find the position function giving the height, s, as a function of the time t.

B. When does the ball hit the ground?

7 EX. #10: An evergreen nursery usually sells a certain shrub after 6 years of growth and shaping. The growth rate during those 6 years is approximated by dh/dt = 1.5t + 5, where t is the time in years, and h is the height in centimeters. The seedlings are 12 centimeters tall when planted (t = 0).

A. Find the height after t years. [Hint: the derivative is a rate of change of a function and the integral is the initial function.]

B. How tall are the shrubs when they are sold?

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