Integration

P1^

6

EXAMPLE 6.8 Evaluate (^) ∫ 12 ()^3142 −+ 4
xx
d.x
SOLUTION
1
2
(^421)
(^242)
31
(^31434)

3

31

∫ ()−+ =−∫ ( + )

= − −−

−−

−−

xx

xx xx

xx

d d

 ++







=− ++







=−()++ −+

4

(^114)

81 1

1

2

(^31)
2
1
8
1
2
x
x x
x
()– ++

4

(^438)
Indefinite integrals
The integral symbol can be used without the limits to denote that a function is to
be integrated. Earlier in the chapter, you saw ddyx
= 2 x ⇒ y = x^2 + c.
An alternative way of expressing this is

∫^2 x^ dx^ =^ x^2 +^ c.
EXAMPLE 6.9 Find (^) ∫(2x^3 − 3 x + 4) dx.
SOLUTION
∫(2x^3 −^3 x^ +^ 4)^ dx^ =+ +
=+ +

24 32 4

2

3

2 4

42

42

xx

xc

xx

xc

–

–.

EXAMPLE 6.10 Find the indefinite integral xx x

(^32)
∫()+ d.
SOLUTION
xx xx xx
xx c
(^323212)
(^25232)
5
2
3

()+ =+()

=+ +

∫∫d d

Read as ‘the

integral of 2x with

respect to x’.

3

2 1

5 2 5 2 2 5

+= , and dividing by

3

2 1

5 2 5 2 2 5

+=

is

the same as multiplying by

3

2 1

5 2 5 2 2 5

+=

.