Cambridge Additional Mathematics

Integration (Chapter 15) 427

2 Integrate with respect tox:

a 3 sinx¡ 2 b 4 x¡2 cosx c sinx¡2 cosx+ex

d x^2

p

x¡10 sinx e

x(x¡1)

3

+ cosx f ¡sinx+2

p

x

3 Find:

a

Z

(x^2 +3x¡2)dx b

Z μ

p

x¡

1

p

x

¶

dx c

Z ³

2 ex¡

1

x^2

́

dx

d

Z

1 ¡ 4 x

x

p

x

dx e

Z

(2x+1)^2 dx f

Z ³

x+

1

x

́ 2

dx

g

Z

2 x¡ 1

p

x

dx h

Z

x^2 ¡ 4 x+10

x^2

p

x

dx i

Z

(x+1)^3 dx

4 Find:

a

Z ¡

p

x+^12 cosx

¢

dx b

Z

(2et¡4 sint)dt c

Z

(3 cost¡sint)dt

5 Findyif:

a dy

dx

=6 b dy

dx

=4x^2 c dy

dx

=5

p

x¡x^2

d

dy

dx

=

1

x^2

e

dy

dx

=2ex¡ 5 f

dy

dx

=4x^3 +3x^2

6 Find f(x) if:

a f^0 (x)=(1¡ 2 x)^2 b f^0 (x)=

p

x¡

2

p

x

c f^0 (x)=

x^2 ¡ 5

x^2

PARTICULAR VALUES

We can find the constant of integrationcif we are given a particular value of the function.

Example 9 Self Tutor

Find f(x) given that:

a f^0 (x)=x^3 ¡ 2 x^2 +3 and f(0) = 2 b f^0 (x) = 2 sinx¡

p

x and f(0) = 4.

a Since f^0 (x)=x^3 ¡ 2 x^2 +3,

f(x)=

Z

(x^3 ¡ 2 x^2 +3)dx

) f(x)=

x^4

4

¡

2 x^3

3

+3x+c

But f(0) = 2,soc=2

Thus f(x)=

x^4

4

¡

2 x^3

3

+3x+2

b f(x)=

Z μ

2 sinx¡x

1

2

¶

dx

) f(x)=2£(¡cosx)¡

x

3

2

3

2

+c

) f(x)=¡2 cosx¡^23 x

3

(^2) +c
But f(0) = 4,
so ¡2 cos 0¡0+c=4
) c=6
Thus f(x)=¡2 cosx¡^23 x
3
(^2) +6.
4037 Cambridge
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Y:\HAESE\CAM4037\CamAdd_15\427CamAdd_15.cdr Monday, 7 April 2014 3:59:18 PM BRIAN