326 Basic Engineering Mathematics
(b) When a sum of several terms is integrated the
result is the sum of the integrals of the separate
terms. For example,
(3x + 2x
2
− 5) dx
=
3x dx +
2x
2 dx −
5 dx
=
3x
2
2
+
2x
3
3
− 5x + c
35.3 Standard integrals
From Chapter 34,
d
dx
(sin ax) = a cos ax. Since integration is the reverse process of differentiation, it
follows that
a cos ax dx = sin ax + c
or
cos ax dx =
1
a
sin ax + c
By similar reasoning
sin ax dx = −
1
a
cos ax + c
e
ax dx =
1
a
e
ax
+ c
and
1
x
dx = ln x + c
From above,
ax n dx =
ax n+1
n + 1
+ c except when
n = −1
When n = −1,
x −1 dx =
1
x
dx = ln x + c
A list of standard integrals is summarized in
Table 35.1.
Table 35.1 Standard integrals
y
y dx
1.
ax n
ax n+1
n + 1
+ c (except when n = −1)
2.
cos ax dx
1
a
sin ax + c
3.
sin ax dx −
1
a
cos ax + c
4.
e ax dx
1
a
e ax + c
5.
1
x
dx
ln x + c
Problem 1. Determine
7x
2 dx
The standard integral,
ax
n dx =
ax n+1
n + 1
+ c
When a = 7 and n = 2,
7x
2 dx =
7x 2+1
2 + 1
+ c =
7x 3
3
+ c or
7
3
x
3
+ c
Problem 2. Determine
2t 3 dt
When a = 2 and n = 3,
2t
3 dt =
2t
3+1
3 + 1
+ c =
2t
4
4
+ c =
1
2
t
4
+ c
Note that each of the results in worked examples 1 and
2 may be checked by differentiating them.
Problem 3. Determine
8 dx
8 dx is the same as
8x 0 dx and, using the general
rule when a = 8 and n = 0, gives
8x
0 dx =
8x 0+1
0 + 1
+ c = 8x + c
In general, if k is a constant then
kdx = kx + c.
Problem 4. Determine
2x dx
When a = 2 and n = 1,
2x dx =
2x
1 dx =
2x 1+1
1 + 1
+ c =
2x 2
2
+ c
= x
2
+ c
Problem 5. Determine
3 +
2
5
x − 6x
2
dx
3 +
2
5
x − 6x 2
dx may be written as
3 dx+
2
5
x dx −
6x 2 dx
i.e., each term is integrated separately. (This splitting
up of terms only applies, however, for addition and