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

x31 c 3 1 x4 = c 4

2

c)

5 2 x dx

2 x51 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

x3 3x2 dx x1 x2 = 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