Integration

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ADDITIONAL MATHEMATICS FORM 5
MODULE 4

INTEGRATION

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CHAPTER 3 : INTEGRATION

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4.1 Integration of Algebraic Functions Exercise A 4.2 The Equation of a Curve from Functions of Gradients. Exercise B SPM Question Assessment Answer

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Indefinite Integral
a)

ò ò a x n a dx = ax + c. xn+ 1 + c. n+ 1

b)

x n dx =

c

)

ò

d

x

=

a

ò

x

n

d

x

=

a n

x +

n

+

1

1

+

c

.

Integration of Algebraic Functions

e) ) The [f (x) ± g(x) ]dx = ò f (x) dx ± d ò Equation of a Curve from Functions of Gradients

ò g(x)dx

y = y =

ò

f '( x ) d x c,

f (x) +

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INTEGRATION

1. Integration is the reverse process of differentiation. dy 2. If y is a function of x and = f '( x) then ò f '( x)dx = y + c, c = constant. dx If

dy = f ( x ), then dx

ò f ( x)dx =

y

4.1.

Integration of Algebraic Functions

Indefinite Integral

a)
b)

ò ò a dx = ax + c. n a and c are constants

xn+ 1 x dx = + c. n+ 1 n c is constant, n is an integer and n ≠ -

c)

ò ax

dx = a ò

ax n + 1 x dx = + c. n+ 1 n a and c are constants n is an

d)

ò [f ( x ) ±

g ( x ) ]dx =

ò

f ( x) dx ±

ò g ( x)dx

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Find the indefinite integral for each of the following. a )  5dx b)  x 3 dx c)

 2 x dx
5

d)

 ( x  3x

2

)dx

Always remember to include ‘+c’ in your answers of indefinite integrals.

Solution : a)

 5dx  5x  c

b)

3  x dx 

x31 c 3 1 x4 = c 4
2

c)

5  2 x dx 

2 x51 c 5 1 2 x6 = c 6 1 = x6  c 3

d)

 ( x  3x

)dx   xdx   3x 2 dx =