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Subject : Mathematics T

Chapter : 9 Integration (1)

9.1 Indefinite Integrals

Notes:-

(a) Standard integrals

(i) a dx ax c

(ii)1

1

nn ax

ax dxn

+c

1. Determine the followin

(a) 5x dx

(c) 3x dx

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Class :

Lesson no :

Date :

Time :

integrals.

(b)

1

2x dx

(d)4

1dx

x

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Cheng Wui Leap

U6 Maths

9

10/2/14(Tue)

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2. Find the following integrals

(a) 2 34 5x dx (b) 2 2x x dx

(c) 1x

dxx

(d)

24 3x dx

(e)

23

x dxx

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Lesson 9 -U6 Maths 3

9.2 Integral of n

ax b

Note:-

1

1

n

n ax bax b dx c

n a

Exercise

Find the integral of the following.

1. 5

4 2x dx

2. 1

21

3 8

3

dx

3. 3

4 5 2dx

x

9.3 Integral of Exponential functions

Notes:-

x xe dx e c

1ax b ax be dx e ca

' f x f xf x e dx e c

Exercise

1. Find

(a) 3 1x

e dx

(b) 2

3xe dx

(c)

21xx

e dx

e

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2. Find

(a)32 x

x e dx

(b)xe

dxx

(c) 2 tan 2sec 2 xe dx

(d)

1

2

xedx

9.4 Integrals of the form

'ln | |

f xdx f x c

f x

Exercise

1. 1

3 1dx

x

2. 3

1dx

x

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3.2

1

2 7

xdx

x

4. cot 4 dx

5. tan 22

dx

6.1

x

x

edx

e

7.3

31

x

x

edx

e

8. 1

1

xdx

x

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Lesson 9 -U6 Maths 6

9.3

21

xdx

9.5 Integrals of Trigonometric Functions

1

cos sinax b dx ax b ca

1sin cosax b dx ax b c

a

2

1sec tanax b dx ax b c

a

2 1

csc cotax b dx ax b ca

sec tan secx x dx x c cosec cot cosecx xdx x c

Exercise

1. 2sinxdx

2. 23sec dx

3.3

2

cos 1

cos

xdx

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4.3

2

3 sin

sin

xdx

x

5. sec tan secx x x dx

6. cot tan cosecx x x dx

7. 2sec cos tanx x x dx

9.6 Techniques of Integration

(i) Use of formula

1

'1

n

n f xf x f x dx c

n

, find the following integrals.

(a) 5

4x dx

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(b) 2

2 33 1x x dx

(c) 2 1x dx

(d)2

3 1

xdx

x

(e) lnx

dxx

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(f) 1x xe e dx

Integration as anti-differentiation

1. Differentiate 2ln 1x x with respect tox. Hence, find2

1

1dx

x .

[Ans. 221

, ln 11

x cx

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2. Given that 3 2 3x x , show that 3

.2 3

dy x

dx x

Hence, find

2 3

xdx

x .

[Ans. 1

33

x 2 3x c ]

3. Find cosd

xdx

. Hence, evaluate cos sinx x x dx .[Ans. cos sin ; cosx x x x x c ]

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9.7 Integration using partial fractions

1. Find2

1

4dx

x [Ans.

1 2ln

4 2

xc

x

]

2. Find2

1

9 16dx

x [Ans.

1 3 4ln

24 3 4

xc

x

]

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3. Find

2

4 1

xdx

x x

[Ans.

2

4ln

1

xc

x

]

4. Find2

7

2 3 2

xdx

x

[Ans.

3ln 2 1 ln 2

2x x c ]