Note Addmath

Teaching and learning module Additional mathematics form 5

CHAPTER 3 NAME:…………………………………………………. FORM :…………………………………………………

Date received : ……………………………… Date completed …………………………. Marks of the Topical Test : ……………………………..

Prepared by : Addational Mathematics Department Sek Men Sains Muzaffar Syah Melaka For Internal Circulations Only

Formulae a) Area under a curve = 5

∫x

a

b

dy

Volume generated =

∫ πy

a

b

2

dx

or

=

∫ πx

a

b

2

dy

Students will be able to: 1. Understand and use the concept of indefinite integral. 1.1 Determine integrals by reversing differentiation. n 1.2 Determine integrals of ax , where a is a constant and n is an integer, n ≠ − 1 . 1.3 Determine integrals of algebraic expressions. 1.4 Find constant of integration, c , in indefinite integrals. 1.5 Determine equations of curves from functions of gradients.

n

1.6 Determine by substitution the integrals of the form (ax + b ) , where an integer and n ≠ − 1 .

a and b are constants, n is

1.1

Determining integrals by reversing differentiation.

a) Integration is the inverse process of differentiation . b) The process of obtaining obtaining y from

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

dy is known as integration. dx

dy = f ′ ( x) dx

c) Integration of y with respect to x , is denoted by d) If y is a function of x and

∫ f ( x)dx then ∫ f ′( x ) dx = y + c

where c is arbitrary constant

1.2 Determining integrals of ax , where integrals of algebraic expressions. Formula : 1. Integral of a constant 2. Integral of ax

n

n

a is a constant and n is an integer, n ≠ − 1 and determining

∫ kdx = kx + c

where c is a constant

( n integer, n ≠ -1) is given by

∫

ax n dx =

ax n +1 +c n +1

Example 2 . Integrate each of the following with respect to x a) ∫ 8dx = b) ∫ − 5dx = c) ∫ x 6 dx

6

d) ∫

1 dx x5

e) ∫ −

3 2x 4

f) ∫

3 x

dx

g) ∫ x

h)...