integration

370 INTEGRAL CALCULUS
(a) From Table 37.1(iv),
∫
( 1
7 sec 2 4t dt = (7) tan 4t + c
4)
= 7 tan 4t + c
4
(b) From Table 37.1(v),
∫
(
3 cosec 2 2θ dθ = (3) − 1 )
cot 2θ + c
2
Problem 10. Determine
∫
∫
(a) 5e 3x dx (b)
= − 3 cot 2θ + c
2
2
dt.
3e4t (a) From Table 37.1(viii),
∫
( 1
5e 3x dx = (5) e
3)
3x + c = 5 3 e3x + c
(b)
∫ ∫ ( )(
2 2 2
3e 4t dt = 3 e−4t dt = − 1 )
e −4t + c
3 4
=− 1 6 e−4t + c = − 1
6e 4t + c
Problem 11. Determine
∫ ∫ ( 3 2m 2 )
(a)
5x dx (b) + 1
dm.
m
(a)
(b)
∫ 3
5x dx = ∫ ( 3
5
∫ ( 2m 2 + 1
m
)( ) 1
dx = 3 x 5 ln x + c
(from Table 37.1(ix))
) ∫ ( 2m
2
dm =
m
=
+ 1 m
∫ (
2m + 1 m
)
dm
)
dm
= 2m2
2 + ln m + c
= m 2 + ln m + c
Now try the following exercise.
Exercise 146 Further problems on standard
integrals
In Problems 1 to 12, determine the indefinite
integrals.
∫ ∫
1. (a) 4dx (b) 7x dx
[(a) 4x + c (b) 7x2
2 + c ]
∫ ∫ 2 5
2. (a)
5 x2 dx (b)
6 x3 dx
[
(a) 2 15 x3 + c (b) 5 ]
24 x4 + c
∫( 3x 2 ) ∫
− 5x
3. (a)
dx (b) (2 + θ) 2 dθ
x
⎡
⎤
(a) 3x2
⎢ 2 − 5x + c
⎥
⎣
(b) 4θ + 2θ 2 + θ3 ⎦
3 + c
∫ ∫ 4
3
4. (a)
3x 2 dx (b) 4x 4 dx
[
(a) −4
]
3x + c −1
(b)
4x 3 + c
∫ √x
∫ 1 √
5. (a) 2 3 4
dx (b) x
4
5 dx
[
(a) 4 √
x
5
5 + c (b) 1 ]
4√
x
9
9 + c
6. (a)
∫ −5
√
t 3 dt
(b) ∫
3
7 5√ x 4 dx
[
(a) 10 √ t
+ c (b) 15 7
∫
∫
7. (a) 3 cos 2x dx (b) 7 sin 3θ dθ
5√ x + c
]
⎡
(a) 3 ⎤
sin 2x + c
⎢ 2 ⎣
(b) − 7 ⎥
3 cos 3θ + c ⎦