4.1 Antiderivatives and Indefinite Integration

23)( xxf

3)( xxF

Suppose you were asked to find a function F whose derivative is

From your knowledge of derivatives, you would probably say

The function F is an antiderivative of f. In general, a function F is an antiderivative of f (x) if

xxfxF )()('

Note that F is an antiderivative, not the antiderivative

Ex:

CxF )(

)(xf

In general:

is the antiderivative of

Example 1

xxf 2)( Example 2: Solving a Differential Equation

Gives the entire family of

antiderivatives

2'y

)(xfdx

dy

dxxfdy )(

Notation for Antiderivatives:When solving a differential

equation

it is convenient to write the differential form

CxFdxxfy )()(

The operation of finding all solutions of this equation is antidifferentiation or indefinite integration

Integrand

Variable of integration

Constant of integration

Practice

dxx3

1 dtt22 1

xdx3 dxx

Practice

dxxxx 24 53

dxx

x2cos

sin

dxx

x 1

xdxsin2

Initial Conditions and Particular Solutions

Solve the differential equation:

13 2 xdx

dy

Solve the differential equation above if the curve passes through (2,4)—called an initial condition.

Initial Conditions and Particular Solutions

Find the general solution of:

0,1

)('2

xx

xF

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

A Vertical Motion Problem

A ball is thrown upward with an initial velocity of 64 feet per second from an initial height of 80 feet, as shown in the figure.

1. Find the position function giving the height s as a function of time t2. When does the ball hit the ground?