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) = 2x 2 then
f (x) = 4x. Thus, the integral of 4x is 2x 2 ; i.e., integration is the process of moving from f (x) to f (x). By
similar reasoning, the integral of 2t is t
2 .
Integration is a process of summation or adding parts
together and an elongated S, shown as
, is used to
replace the words ‘the integral of’. Hence, from above,
4x = 2x 2 and
2t is t 2 .
In differentiation, the differential coefficient
dy
dx
indicates that a function of x is being differentiated with
respect to x, the dx indicating that it is ‘with respect
to x’.
In integration the variable of integration is shown
by adding d (the variable) after the function to be
integrated. Thus,
4x dx means ‘the integral of 4x with respect to x’,
and
2t dt means ‘the integral of 2t with respect to t ’
As stated above, the differential coefficient of 2x 2 is 4x,
hence;
4x dx = 2x 2 . However, the differential coefficient of 2x 2 + 7 is also 4x. Hence,
4x dx could also
be equal to 2x 2 + 7. To allow for the possible presence
of a constant, whenever the process of integration is
performed a constant c is added to the result. Thus,
4x dx = 2x
2
+ c and
2t dt = t
2
+ c
c is called the arbitrary constant of integration.
35.2 The general solution of integrals
of the form ax n
The general solution of integrals of the form
ax n dx,
where a and n are constants and n = −1 is given by
ax
n dx =
ax
n+1
n + 1
+ c
Using this rule gives
(i)
3x
4 dx =
3x 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 =
2x
−2 dx =
2x −2+1
−2 + 1
+ c
=
2x −1
−1
+ c = −
2
x
+ c
(iv)
√
x dx =
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 differentiation.
(a) The integral of a constant k is kx + c. For
example,
8 dx = 8x + c and
5 dt = 5t + c
DOI: 10.1016/B978-1-85617-697-2.00035-1
Introduction to integration
35.1 The process of integration
The process of integration reverses the process of
differentiation. In differentiation, if f (x) = 2x 2 then
f (x) = 4x. Thus, the integral of 4x is 2x 2 ; i.e., integration is the process of moving from f (x) to f (x). By
similar reasoning, the integral of 2t is t
2 .
Integration is a process of summation or adding parts
together and an elongated S, shown as
, is used to
replace the words ‘the integral of’. Hence, from above,
4x = 2x 2 and
2t is t 2 .
In differentiation, the differential coefficient
dy
dx
indicates that a function of x is being differentiated with
respect to x, the dx indicating that it is ‘with respect
to x’.
In integration the variable of integration is shown
by adding d (the variable) after the function to be
integrated. Thus,
4x dx means ‘the integral of 4x with respect to x’,
and
2t dt means ‘the integral of 2t with respect to t ’
As stated above, the differential coefficient of 2x 2 is 4x,
hence;
4x dx = 2x 2 . However, the differential coefficient of 2x 2 + 7 is also 4x. Hence,
4x dx could also
be equal to 2x 2 + 7. To allow for the possible presence
of a constant, whenever the process of integration is
performed a constant c is added to the result. Thus,
4x dx = 2x
2
+ c and
2t dt = t
2
+ c
c is called the arbitrary constant of integration.
35.2 The general solution of integrals
of the form ax n
The general solution of integrals of the form
ax n dx,
where a and n are constants and n = −1 is given by
ax
n dx =
ax
n+1
n + 1
+ c
Using this rule gives
(i)
3x
4 dx =
3x 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 =
2x
−2 dx =
2x −2+1
−2 + 1
+ c
=
2x −1
−1
+ c = −
2
x
+ c
(iv)
√
x dx =
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 differentiation.
(a) The integral of a constant k is kx + c. For
example,
8 dx = 8x + c and
5 dt = 5t + c
DOI: 10.1016/B978-1-85617-697-2.00035-1