Cambridge Additional Mathematics

Integration (Chapter 15) 423

Example 5 Self Tutor

If y=x^4 +2x^3 , find

dy

dx

. Hence find

Z

(2x^3 +3x^2 )dx.

If y=x^4 +2x^3 then dy

dx

=4x^3 +6x^2

)

Z

(4x^3 +6x^2 )dx=x^4 +2x^3 +c

)

Z

2(2x^3 +3x^2 )dx=x^4 +2x^3 +c

) 2

Z

(2x^3 +3x^2 )dx=x^4 +2x^3 +c

)

Z

(2x^3 +3x^2 )dx=^12 x^4 +x^3 +c

EXERCISE 15D

1 If y=x^7 , find

dy

dx

. Hence find

Z

x^6 dx.

2 If y=x^3 +x^2 , find

dy

dx

. Hence find

Z

(3x^2 +2x)dx.

3 If y=e^2 x+1, find

dy

dx

. Hence find

Z

e^2 x+1dx.

4 If y=(2x+1)^4 find

dy

dx

. Hence find

Z

(2x+1)^3 dx.

Example 6 Self Tutor

Suppose y=

p

5 x¡ 1.

a Find

dy

dx

. b Hence find

Z

1

p

5 x¡ 1

dx.

a y=

p

5 x¡ 1

=(5x¡1)

1

2

)

dy

dx

=^12 (5x¡1)

¡^12

(5) fchain ruleg

=

5

2

p

5 x¡ 1

b Usinga,

Z

5

2

p

5 x¡ 1

dx=

p

5 x¡1+c

)^52

Z

1

p

5 x¡ 1

dx=

p

5 x¡1+c

)

Z

1

p

5 x¡ 1

dx=^25

p

5 x¡1+c

5 If y=x

p

x, find

dy

dx

. Hence find

Z

p

xdx.

6 If y=

1

p

x

, find

dy

dx

. Hence find

Z

1

x

p

x

dx.

crepresents a general

constant, so is simply any

value c 2 R.

Instead of writing c 2 ,we

can therefore still write

justc.

4037 Cambridge

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Y:\HAESE\CAM4037\CamAdd_15\423CamAdd_15.cdr Monday, 7 April 2014 4:03:33 PM BRIAN