Cambridge Additional Mathematics

432 Integration (Chapter 15)

Example 13 Self Tutor

Find:

a

Z 3

1

(x^2 +2)dx b

Z ¼

3

0

sinxdx

a

Z 3

1

(x^2 +2)dx

=

·

x^3

3

+2x

̧ 3

1

=

³

33

3 + 2(3)

́

¡

³

13

3 + 2(1)

́

=(9+6)¡(^13 +2)

=12^23

b

Z ¼

3

0

sinxdx

=[¡cosx]

¼

3

0

=(¡cos¼ 3 )¡(¡cos 0)

=¡^12 +1

=^12

EXERCISE 15G

Use questions 1 to 4 to check the properties of definite integrals.

1 Find: a

Z 4

1

p

xdxand

Z 4

1

(¡

p

x)dx b

Z 1

0

x^7 dx and

Z 1

0

(¡x^7 )dx

2 Find: a

Z 1

0

x^2 dx b

Z 2

1

x^2 dx c

Z 2

0

x^2 dx d

Z 1

0

3 x^2 dx

3 Find: a

Z 2

0

(x^3 ¡ 4 x)dx b

Z 3

2

(x^3 ¡ 4 x)dx c

Z 3

0

(x^3 ¡ 4 x)dx

4 Find: a

Z 1

0

x^2 dx b

Z 1

0

p

xdx c

Z 1

0

(x^2 +

p

x)dx

5 Evaluate:

a

Z 1

0

x^3 dx b

Z 2

0

(x^2 ¡x)dx c

Z 1

0

exdx

d

Z ¼

6

0

cosxdx e

Z 4

1

μ

x¡

3

p

x

¶

dx f

Z 9

4

x¡ 3

p

x

dx

g

Z 3

1

1

x

dx h

Z¼

2

¼

3

sinxdx i

Z 2

1

(e¡x+1)^2 dx

j

Z 6

2

1

p

2 x¡ 3

dx k

Z 1

0

e^1 ¡xdx l

Z ¼

6

0

sin(3x)dx

6 Findmsuch that

Z 2 m

m

(2x¡1)dx=4.

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Additional Mathematics
Y:\HAESE\CAM4037\CamAdd_15\432CamAdd_15.cdr Monday, 7 April 2014 3:59:55 PM BRIAN