Unit 3 INTEGRATION (Polynomial) R

INTEGRATION

UNIT 3: INTEGRATION

INTRODUCTION

Every operation in mathematics has its inverse. In this chapter we learn how to reverse
the process of differentiation with the process of integration.

LEARNING OUTCOMES

Upon completion of this unit, students should be able to:

1. find the integrals of functions by considering integration as a reverse of
differentiation.

2. integrate indefinite integrals by rule of integration.
3. integrate composite polynomial functions by using substitution method.
4. integrate product of functions by part.
5. integrate quotient of functions using partial fraction method.
6. evaluate the definite integrals.

1 DUM 20032

INTEGRATION

3.1 INTEGRATION AS THE REVERSE OF DIFFERENTIATION

dy
Integration is the reverse process of differentiation. The process of obtaining

dx
from y (a function of a) is known as differentiation. Hence, the process of

dy
obtaining y from is known as integration. Integration of y with respect to x, is

dx

 denoted by f(x) dx . The symbols f(x) dx denote the integral of f(x) with

respect to the variable x. For example:

y = x3 + c Differentiation dy = x2
3 Integration dx

3.2 INDEFINITE INTEGRAL

From which the derivative 5x4 was derived?. Take a look at these examples:

( )d x5 = 5x4  5x 4 dx = x5
 5x 4 dx = x5 + 3
dx  5x 4 dx = x5 − 1

( )also d x5 + 3 = 5x4
dx

( )and d x5 − 1 = 5x4
dx

Any constant terms in the original expression becomes zero in the derivative.
Therefore the presence of such constant term is replaced by adding a symbol c
to the result of integration.

 f'(x) dx = f(x) + c

c is known as the constant of integration and must always be included. Such
integral is called an indefinite integral.

2 DUM 20032

INTEGRATION

3.1.1 INTEGRATION OF POLYNOMIALS

1.  a dx = ax + c where a is aconstant 4. (ax + b)n dx = (ax + b)n + 1 +c

2. xn dx = xn + 1 + c (n + 1) d (ax + b)
(n + 1) dx
(n  −1)
; (n  − 1)

3. axn dx = a xn + 1 + c (n  −1)   5. f(x)  g(x) dx = f(x) dx  g(x) dx
(n + 1)

Example 3.1 : Integrating a constant

Integrate the following with respect to x :

(a) 2 (b) −5 (c) 1
3

Solution :

(a) 2 dx = 2x + c

(b) − 5 dx = −5x + c
(c) 1 dx = 1 x + c

33

3 DUM 20032

INTEGRATION

Example 3.2 : Integration of xn

Integrate the following with respect to x :

(a) x 5 (b) 1 (c) 1
x3 x

Solution :  x 5 dx = 1 x 5+1 + c = 1 x 6 + c
(a) 5 +1 6
(b)
 1 dx = x −3 dx = − 1 x −2 + c = − 1 + c
(c) x 3 2 2x 2

 1 −1 1 − 1 +1 1

dx = x 2 dx = x 2 + c = 2x 2 + c = 2 x + c
x − 1 +1
2

4 DUM 20032

INTEGRATION

Example 3.3 : Integration of axn

Find the following integrals:

(a) -4x2 (b) 2x3 (c) 3x5

Solution :  − 4x 2 dx = −4 x2 dx = −4  x3  + c = − 4 x 3 +c
(a) 3 3
(b)
(c)  2x 3 dx = 2 x3 dx = 2  x4  + c = 1 x4 +c
4 2

 3x 5 dx = 3 x5 dx = 3  x6  + c = 1 x 6 +c
6 2

5 DUM 20032

INTEGRATION

Example 3.4 : Integral of (ax + b)n

Find: (b)  (3 − 6x)−4 dx (c)  (1− x)− 1 dx
2
(a)  (2x + 3)4 dx

Solution :

(a)  (2x + 3)4 dx = 1 (2x + 3)5 + c = 1 (2x + 3)5 + c

(5)(2) 10

(b)  (3 − 6x)−4 dx = ( −3 1 −6 ) ( 3 − 6x )−3 +c =1 (3 − 6x)−3 +c
18
)(

(c) (1− x )− 1 dx = 1 11
2
(1− x)2 + c = −2 (1− x)2 + c
 1 
 2  ( −1)

6 DUM 20032

INTEGRATION

Example 3.5 : Integral of sum and differences

Find the following integrals:

(a)   x2 + 3 + 1  dx (b)  (2x −1)2 dx (c)  x+ 1 dx
 x3  x3

Solution :     x 2 + 3 + 1  dx = x 2 dx + 3 1 dx + x −3 dx
(a)  x3 
= 1 x 3 + 3x − 1 x −2 + c
32

  ( )(b) (2x − 1)2 dx = 4x 2 − 4x + 1 dx = 4 x3 − 2x 2 + x + c
3

  ( )(c) x + 1 dx = x −2 + x −3 dx = x −1 + x −2 + c = − 1 − 1 + c
x3 −1 −2 x 2x 2

7 DUM 20032

INTEGRATION

UNIT EXERCISE 3.1

1. Find the following integrals:

(a)  − 15 dx (b)  6 dx
x4
( )(c)  3x2−4x3 dx
(d)   3 − 4 + 6  dx
(e)  (1 + 4x − 6x2 ) dx  x2 x3 

(g)  (3x + 2)(2x −1) dx (f )   x + 1  dx
 x2 

3x3 − 3x 2 + 4
x2
(h)  dx

8 DUM 20032

INTEGRATION

2. Find the following integrals:

(a)  (x + 1)3 dx (b)  (x − 9)8 dx
(c)  (4x + 7)5 dx
(d)  (3x − 8)6 dx
(e) 2(4 − 3x)−6 dx
(f)  (4 5 dx
− 3x)3

9 DUM 20032