4.1 Antiderivatives and Indefinite Integration

Objective: Use indefinite integral notation for antiderivatives and use basic integration rules to find antiderivatives.

Miss BattagliaAP Calculus AB/BC

Definition of Antiderivative

A function F is an antiderivative of f on an interval I if F’(x)=f(x) for all x in I.

Representation of Antiderivatives

If F is an antiderivative of f on an interval I, then G is an antiderivative of f on the interval I if and only if G is of the form G(x)=f(x)+C, for all x in I where C is a constant.

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

Find the general solution of the differential equation y’=2

Solving a Differential Equation

When solving a differential equation of the form

it is convenient to write it in the equivalent differential form

The general solution is denoted by

Notation for Antiderivatives

*Refer to handout

Basic Integration Rules

Describe the antiderivatives of 3x.

Applying the Basic Integration Rules

Original Rewrite Integrate Simplify

Original Integral

Rewrite Integrate Simplify

Rewriting Before Integrating

a. b. c.

Integrating Polynomial Functions

Rewriting Before Integrating

Rewriting Before Integrating

Find the general solution of

and find the particular solution that satisfies the initial condition F(1)=0.

Finding a Particular Solution

A ball is thrown upward with an initial velocity of 64 ft/sec from an initial height of 80 ft.a) Find the position function giving the height s as

a function of the time t.b) When does the ball hit the ground?

Solving a Vertical Motion Problem

Original Integral

Rewrite Integrate Simplify

Rewriting Before Integrating

Read 4.1 Page 255 #5,6,10,11,12,14,27,29,33,35,37,38,49,60,61,64,84

Classwork/Homework