4001 ANTIDERIVATIVES AND INDEFINITE INTEGRATION

BC Calculus

ANTIDERIVATIVES AND INDEFINITE INTEGRATION

 

Rem:

DEFN: A function F is called an Antiderivative of the function f, if for

every x in f: F /(x) = f(x)

If f (x) = then F(x) =

or since

If f / (x) = then f (x) =

2

2

2

3 7

3 12

3

y x x

y x x

y x x

2 3y x

y

y

23x

3x3 2( ) 3

dx x

dx

23x

3x

Notation:

Differential Equation

  Differential Form (REM: A Quantity of change)

 Integral symbol =

 Integrand =

 Variable of Integration =  

( )dy

f xdx

( )dy f x dx

( )y f x dx ( )f x

dx

The Variable of Integration

( )y f x dx

1 22

Gm m

drr

Newton’s Law of gravitational attraction

NOW: dr tells which variable is being integrated r

Will have more meanings later!

ANTIDERIVATIVES

Layman’s Idea:

A) What is the function that has f (x) as its derivative?.

-Power Rule:

-Trig:

B) The antiderivative is never unique, all answers must include a

+ C (constant of integration)

1

( ) ( )1

nn x

f x x F xn

( ) cos ( ) sinf x x F x x ( ) sin ( ) cosf x x F x x

The Family of Functions whose derivative is given.

Verify the statement by showing the derivative of the right side equals the integral of the left side.

4 3

32

9 3

sin( ) cos( )3

dx cx x

tt t dt t c

Family of Graphs +C

cos( )dy

xdx

The Family of Functions whose derivative is given.

Notation:

Differential Equation

  Differential Form

( REM: A Quantity of change)Increment of change

  Antiderivative or Indefinite Integral

Total (Net) change

3 4dy

xdx

dy

y

dy

General Solution

A) Indefinite Integration and the Antiderivative are the same thing.  General Solution _________________________________________________________  ILL:

( ) ( )f x dx F x c 3

3

1

1

2

xdx

dxx x

dxx

General Solution: EX 1.

General Solution: The Family of Functions ( ) ( )f x dx F x c

3

1dx

xEX 1:

General Solution: EX 2.

General Solution: The Family of Functions ( ) ( )f x dx F x c

(2sin )x dxEX 2:

General Solution: EX 3.

General Solution: The Family of Functions ( ) ( )f x dx F x c / 1( )f x

xEX 3:

Careful !!!!!

Special Considerations

4

2

2

3

( 3)

3 1

1

x dx

x dx

x xdx

x

Initial Condition Problems:

B) Initial Condition Problems:Particular solution < the single graph of the Family –

through a given point> ILL: through the point (1,1)

-Find General solution

-Plug in Point < Initial Condition >and solve for C

2 1dy

xdx

through the point (1,1)

Initial Condition Problems: EX 4.

B) Initial Condition Problems:Particular solution < the single graph of the Family –

through a given point.> Ex 4:

/ 1( )

2f x x

11

2f

Initial Condition Problems: EX 5.

B) Initial Condition Problems:Particular solution < the single graph of the Family –

through a given point.> Ex 5:

/ ( ) cos( )f x x ( ) 13

f

Initial Condition Problems: EX 6.

B) Initial Condition Problems:

A particle is moving along the x - axis such that its acceleration is .

At t = 2 its velocity is 5 and its position is 10.

Find the function, , that models the particle’s motion.

( ) 2a t

( )x t

Initial Condition Problems: EX 7.

B) Initial Condition Problems:

EX 7:

If no Initial Conditions are given: 

Find if/// ( ) 1f x ( )f x

Last Update:

• 12/17/10

• Assignment– Xerox