Chapter 35

Introduction to integration

35.1 The process of integration

The process of integration reverses the process of
differentiation. In differentiation, if f(x)= 2 x^2 then
f′(x)= 4 x.Thus, the integral of 4xis 2x^2 ; i.e., inte-
gration is the process of moving fromf′(x)tof(x).By
similar reasoning, the integral of 2tist^2.
Integration is a process of summation or adding parts
together and an elongatedS,shownas

∫
,isusedto
replace the words ‘the integral of’. Hence, from above,∫
4 x= 2 x^2 and

∫

2 tist^2.

In differentiation, the differential coefficient

dy

dx

indi-

cates that a function ofxis being differentiated with
respect tox,thedxindicating that it is ‘with respect
tox’.
In integration the variable of integration is shown
by adding d(the variable) after the function to be
integrated. Thus,

∫

4 xdxmeans ‘the integral of 4xwith respect tox’,

and

∫

2 tdtmeans ‘the integral of 2twith respect tot’

As stated above, the differential coefficient of 2x^2 is 4x,
hence;

∫
4 xdx= 2 x^2. However, the differential coeffi-
cient of 2x^2 +7isalso4x. Hence,

∫
4 xdxcould also
be equal to 2x^2 +7. To allow for the possible presence
of a constant, whenever the process of integration is
performed a constantcis added to the result. Thus,
∫
4 xdx= 2 x^2 +c and

∫

2 tdt=t^2 +c

cis called thearbitrary constant of integration.

35.2 The general solution of integrals

of the formaxn

The general solution of integrals of the form

∫

axndx,

whereaandnare constants andn=−1isgivenby

∫

axndx=

axn+^1

n+ 1

+c

Using this rule gives

(i)

∫

3 x^4 dx=

3 x^4 +^1

4 + 1

+c=

3

5

x^5 +c

(ii)

∫

4

9

t^3 dt dx=

4

9

(

t^3 +^1

3 + 1

)

+c=

4

9

(

t^4

4

)

+c

=

1

9

t^4 +c

(iii)

∫

2

x^2

dx=

∫

2 x−^2 dx=

2 x−^2 +^1

− 2 + 1

+c

=

2 x−^1

− 1

+c=−

2

x

+c

(iv)

∫

√

xdx=

∫

x

1

2 dx=

x

1

2 +^1

1

2 +^1

+c=

x

3

2

3

2

+c

=

2

3

√

x^3 +c

Each of these results may be checked by differentia-

tion.

(a) The integral of a constant k iskx+c.For

example,

∫

8 dx= 8 x+c and

∫

5 dt= 5 t+c

DOI: 10.1016/B978-1-85617-697-2.00035-1