Integration

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ADDITIONAL MATHEMATICS FORM 5
MODULE 4

INTEGRATION

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CHAPTER 3 : INTEGRATION

Content
Concept Map

page
2 3–4 5 6 7 8–9 10 – 11 12

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 = x 2 3 x3  c 2 3 x2 =  x3  c 2

Find the indefinite integral for each of the following. a)



 x  3x  dx
2

x

4

b)

x x

2

4    3  4  dx x  

a)

Solution : x  3x2





x

4

dx 

 x 3x2    x4  x4 dx  

b)

2

4   2 4  3  4  dx =   3x  2  dx x  x    =  3x2  4 x 2 dx  x 1  3x 3 =  4 c 3  1  4 = x3   c x

  x3  3x2 dx  x1  x2 =  3 c 2  1  1 3 = 2  c 2x x









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1. Find

  3x

2

 4 x  10 dx.



[3m]

2. Find

 x

2

 1  2 x  3  dx.



[3m]

1  3. Find   2 x   dx. x 

2

[3m]

4. Find

  2x 



3

x

3   2  dx. 4 x 

[3m]

6x  5 5. Integrate with respect to x. x3

[3m]

6. Find



x

5

 4x2 2x
4

 dx

[3m]

3   7. Find  x  6  6  dx . x  
2

[3m]

8. Integrate

x 2  3x  2 with respect to x. x 1 [3m]

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The Equation of a Curve from Functions of Gradients
dy  f '( x), then the equation of the curve is dx

If the gradient function of the curve is

y 



f '( x ) dx
c is constant.

y  f ( x)  c,

Find the equation of the curve that has the gradient function 3x − 2 and
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