Calculus 2: Antiderivatives - Lesson Notes, Slides of Differential and Integral Calculus

Download Calculus 2: Antiderivatives - Lesson Notes and more Slides Differential and Integral Calculus in PDF only on Docsity!

MATH

Calculus

ANTIDERIVATIVES

OBJECTIVES:

At the end of the lesson the

students are expected to:

know the relationship between

differentiation

and integration;

identify and explain the different parts

of the

integral operation; and

perform basic integration by applying

the power

formula and the properties of the

In general, once any single antiderivative is

known, the other antiderivatives can be obtained

by adding constants to the known derivative.

Thus,

are all antiderivatives of.

2

3

1

5 ,

3

1

2 ,

3

1

,

3

1

3 3 3 3

x x  x  x 

2

f ( x )  x

Theorem If F(x) is any antiderivative of f(x) on an

open interval, then for any constant C the

function F(x)+C is also an antiderivative on that

interval. Moreover, each antiderivative of f(x) on

the interval can be expressed in the form F(x)+C

by choosing the constant C appropriately.

DEFINITION: THE INDEFINITE INTEGRA

The process of finding antiderivatives is

called antidifferentiation or

integration. Thus, if

then integrating (or antidifferentiating) the

function f(x) produces an antiderivative of

the form F(x)+C. To emphasize this

process, we use the following integral

notation

 F ( x ) f ( x )

dx

d



f ( x ) dx  F ( x )  C

Some of the properties of the indefinite integral

and basic integration formulas, which need no

proof from the fact that these properties are

also known properties of differentiation are

listed below.

Properties of Indefinite Integral and Basic

Integration Formula:

; 1

1

.

. [ ( ) ( ) .... ( )] ( ) ( ) ... ( ) . ( ) ( ) ( )

.

1

1 2 1 2 3

  





      

  

 





   

 



C n

n

x

iv x dx

iii f x f x f x dx f x dx f x dx f x dx

ii cf x dx c f x dx cF x C

i dx x C

n

n

n

EXAMPLE

dy y y y a x b dx x x dx x dx

              3 2 3 2 2 2 3 3 2 1

5. 2
6. 3 6 7

2 aluate the following integral.