Cambridge Additional Mathematics

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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

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

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

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