integration

400 INTEGRAL CALCULUS
∫
[ 3θ
sin 4θ
= + sin 2θ +
2 8
[ 3
( π
)
= + sin 2π 2 4 4
= 3π 8 + 1 = 2.178,
Problem 8.
∫
sin 2 t cos 4 t dt =
] π
4
0
+
sin 4(π/4)
8
]
− [0]
correct to 4 significant figures
Find ∫ sin 2 t cos 4 t dt.
sin 2 t(cos 2 t) 2 dt
∫ ( )( ) 1 − cos 2t 1 + cos 2t 2
=
dt
2
2
= 1 ∫
(1 − cos 2t)(1 + 2 cos 2t + cos 2 2t)dt
8
= 1 ∫
(1 + 2 cos 2t + cos 2 2t − cos 2t
8
− 2 cos 2 2t − cos 3 2t)dt
= 1 ∫
(1 + cos 2t − cos 2 2t − cos 3 2t)dt
8
= 1 ∫ [
( ) 1 + cos 4t
1 + cos 2t −
8
2
]
− cos 2t(1 − sin 2 2t) dt
= 1 8
= 1 8
∫ ( )
1 cos 4t
− + cos 2t sin 2 2t dt
2 2
(
)
t sin 4t
− + sin3 2t
+ c
2 8 6
3. 2 sin 3 t cos 2 t [ −2
3 cos3 t + 2 ]
5 cos5 t + c
[
]
4. sin 3 x cos 4 − cos 5 x
x
+ cos7 x
+ c
5 7
5. 2 sin 4 2θ [ 3θ
4 − 1 ]
1
sin 4θ +
4 32 sin 8θ + c
[ t
6. sin 2 t cos 2 t
8 − 1 ]
32 sin 4t + c
40.4 Worked problems on integration
of products of sines and cosines
∫
Problem 9.
Determine ∫ sin 3t cos 2t dt.
sin 3t cos 2t dt
∫ 1
= [sin (3t + 2t) + sin (3t − 2t)] dt,
2
from 6 of Table 40.1, which follows from Section
18.4, page 183,
∫
(sin 5t + sin t)dt
= 1 2
= 1 ( −cos 5t
2 5
)
− cos t + c
∫ 1
Problem 10. Find cos 5x sin 2x dx.
3
Now try the following exercise.
Exercise 157 Further problems on integration
of powers of sines and cosines
In Problems 1 to 6, integrate with respect to the
variable.
[
]
1. sin 3 θ
(a)−cos θ + cos3 θ
+ c
3
[
]
2. 2 cos 3 2x
sin 2x − sin3 2x
+ c
3
∫ 1
cos 5x sin 2x dx
3
= 1 ∫ 1
[sin (5x + 2x) − sin (5x − 2x)] dx,
3 2
from 7 of Table 40.1
= 1 ∫
(sin 7x − sin 3x)dx
6
= 1 ( )
−cos 7x cos 3x
+ + c
6 7 3