Chapter 37
Standard integration
37.1 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 is also 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.
37.2 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 is given by:
ax
n dx =
ax n+1
n + 1
+ c
This rule is true when n is fractional, zero, or a positive
or negative integer, with the exception of n =−1.
Using this rule gives:
(i)
3x
4 dx =
3x 4+1
4 + 1
+ c =
3
5
x
5
+ c
(ii)
2
x 2 dx =
2x
−2 dx =
2x −2+1
−2 +1
+ c
=
2x −1
−1
+ c =
−2
x
+ c, and
(iii)
√
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 three results may be checked by differentiation.
(a) The integral of a constant k is kx + c. For
example,
8 dx = 8x + c
(b) When a sum of several terms is integrated the result
is the sum of the integrals of the separate terms.
Standard integration
37.1 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 is also 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.
37.2 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 is given by:
ax
n dx =
ax n+1
n + 1
+ c
This rule is true when n is fractional, zero, or a positive
or negative integer, with the exception of n =−1.
Using this rule gives:
(i)
3x
4 dx =
3x 4+1
4 + 1
+ c =
3
5
x
5
+ c
(ii)
2
x 2 dx =
2x
−2 dx =
2x −2+1
−2 +1
+ c
=
2x −1
−1
+ c =
−2
x
+ c, and
(iii)
√
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 three results may be checked by differentiation.
(a) The integral of a constant k is kx + c. For
example,
8 dx = 8x + c
(b) When a sum of several terms is integrated the result
is the sum of the integrals of the separate terms.