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