328 Basic Engineering Mathematics
From 3 of Table 35.1,
5 sin 2θdθ = (5)
−
1
2
cos 2θ + c
= −
5
2
cos 2θ+ c
Problem 13. Determine
5e 3x dx
From 4 of Table 35.1,
5e
3x dx = (5)
1
3
e
3x
+ c
=
5
3
e
3x
+ c
Problem 14. Determine
2
3e 4t dt
2
3e 4t dt =
2
3
e
−4t dt
=
2
3
−
1
4
e
−4t
+ c
= −
1
6
e
−4t
+ c = −
1
6e 4t + c
Problem 15. Determine
3
5x
dx
From 5 of Table 35.1,
3
5x
dx =
3
5
1
x
dx
=
3
5
ln x + c
Problem 16. Determine
2x 2 + 1
x
dx
2x 2 + 1
x
dx =
2x 2
x
+
1
x
dx
=
2x +
1
x
dx =
2x 2
2
+ ln x + c
= x
2
+ ln x + c
Now try the following Practice Exercise
Practice Exercise 139 Standard integrals
(answers on page 355)
Determine the following integrals.
1. (a)
4 dx
(b)
7x dx
2. (a)
5x 3 dx
(b)
3 t 7 dt
3. (a)
2
5
x
2 dx
(b)
5
6
x
3 dx
4. (a)
(2x 4 − 3x) dx
(b)
(2 − 3t 3 ) dt
5. (a)
3x 2 − 5x
x
dx (b)
(2 + θ) 2 dθ]
6. (a)
(2 + θ)(3θ − 1) dθ
(b)
(3x − 2)(x 2 + 1)dx
7. (a)
4
3x 2 dx
(b)
3
4x 4 dx
8. (a) 2
√
x 3 dx
(b)
1
4
4
√
x 5 dx
9. (a)
−5
√
t 3
dt
(b)
3
7
5
√
x 4
dx
10. (a)
3 cos2x dx
(b)
7 sin3θ dθ
11. (a)
3 sin
1
2
x dx
(b)
6 cos
1
3
x dx
12. (a)
3
4
e 2x dx
(b)
2
3
dx
e 5x
13. (a)
2
3x
dx
(b)
u 2 − 1
u
du
14. (a)
(2 + 3x) 2
√
x
dx
(b)
1
t
+ 2t
2
dt
35.4 Definite integrals
Integrals containing an arbitrary constant c in their
results are called indefinite integrals since their precise
value cannot be determined without further information.
Definite integrals are those in which limits are applied.
If an expression is written as [x] b
a , b is called the upper
limit and a the lower limit. The operation of applying
the limits is defined as [x] b
a = (b) − (a).