Download unit 12.7 - integration 7

“JUST THE MATHS”
UNIT NUMBER
12.7
INTEGRATION 7
(Further trigonometric functions)
by
A.J.Hobson
12.7.1
12.7.2
12.7.3
12.7.4
Products of sines and cosines
Powers of sines and cosines
Exercises
Answers to exercises
UNIT 12.7 - INTEGRATION 7 - FURTHER TRIGONOMETRIC FUNCTIONS
12.7.1 PRODUCTS OF SINES AND COSINES
In order to integrate the product of a sine and a cosine, or two cosines, or two sines, we may
use one of the following trigonometric identities:
1
[sin(A + B) + sin(A − B)] ;
2
1
cosAsinB ≡ [sin(A + B) − sin(A − B)] ;
2
1
cosAcosB ≡ [cos(A + B) + cos(A − B)] ;
2
1
sinAsinB ≡ [cos(A − B) − cos(A + B)] .
2
sinAcosB ≡
EXAMPLES
1. Determine the indefinite integral
Z
sin 2x cos 5x dx.
Solution
Z
1Z
[sin 7x − sin 3x] dx
sin 2x cos 5x dx =
2
=−
cos 7x cos 3x
+
+ C.
14
6
2. Determine the indefinite integral
Z
sin 3x sin x dx.
Solution
Z
1Z
sin 3x sin x dx =
[cos 2x − cos 4x] dx
2
=
sin 2x sin 4x
−
+ C.
4
8
1
12.7.2 POWERS OF SINES AND COSINES
In this section, we consider the two integrals,
Z
n
sin x dx and
Z
cosn dx,
where n is a positive integer.
(a) The Complex Number Method
A single method which will cover both of the above integrals requires us to use the methods
of Unit 6.5 in order to express cosn x and sinn x as a sum of whole multiples of sines or cosines
of whole multiples of x.
EXAMPLE
Determine the indefinite integral
Z
sin4 x dx.
Solution
By the complex number method,
1
sin4 x ≡ [cos 4x − 4 cos 2x + 3].
8
The Working:
1
j 2 sin x ≡ z −
z
4 4
4
4
,
where z ≡ cos x + j sin x.
That is,
2
1
1
16sin x ≡ z − 4z . + 6z 2 .
z
z
4
4
3
or, after cancelling common factors,
2
3
1
− 4z.
z
4
+
1
z
;
4
1
1
1
16sin x ≡ z − 4z + 6 − 2 + 4 ≡ z 4 + 4 − 4 z 2 + 2 + 6,
z
z
z
z
4
4
2
which gives
16sin4 x ≡ 2 cos 4x − 8 cos 2x + 6,
or
1
sin4 x ≡ [cos 4x − 4 cos 2x + 3].
8
Hence,
Z
1 sin 4x
sin 2x
sin x dx =
−4
+ 3x + C
8
4
2
4
=
1
[sin 4x − 8 sin 2x + 12x] + C.
32
(b) Odd Powers of Sines and Cosines
The following method uses the facts that
d
d
[sin x] = cos x and
[cos x] = sin x.
dx
dx
We illustrate with examples in which use is made of the trigonometric identity
cos2 A + sin2 A ≡ 1.
EXAMPLES
1. Determine the indefinite integral
Z
sin3 x dx.
3
Solution
Z
sin3 x dx =
Z
sin2 x. sin x dx.
1 − cos2 x sin x dx
That is,
Z
3
sin xdx =
=
Z
Z
sin x − cos2 x. sin x dx
= − cos x +
cos3 x
+ C.
3
2. Determine the indefinite integral
Z
cos7 x dx.
Solution
Z
cos7 x dx =
Z
cos6 x. cos x dx.
3
That is,
=
Z
cos7 x dx =
Z
Z
1 − sin2 x . cos x dx
1 − 3sin2 x + 3sin4 x − sin6 x . cos x dx
= sin x − sin3 x + 3.
sin5 x sin7 x
−
+ C.
5
7
(c) Even Powers of Sines and Cosines
The method illustrated here becomes tedious if the even power is higher than 4. In such
cases, it is best to use the complex number method in paragraph (a) above.
In the examples which follow, we shall need the trigonometric identity
cos 2A ≡ 1 − 2sin2 A ≡ 2cos2 A − 1.
4
EXAMPLES
1. Determine the indefinite integral
Z
sin2 x dx.
Solution
Z
2
sin x dx =
1
x sin 2x
(1 − cos 2x) dx = −
+ C.
2
2
4
Z
2. Evaluate the definite integral
π
2
Z
cos2 x dx.
0
Solution
Z
π
2
cos2 x dx =
π
2
Z
0
0
1
(1 + cos 2x) dx
2
x sin 2x
=
+
2
4
π2
=
0
π
.
4
3. Determine the indefinite integral
Z
cos4 x dx.
Solution
Z
4
cos x dx =
Z
h
2
cos x
i2
dx =
Z
2
1
(1 + cos 2x)
2
That is,
Z
cos4 x dx =
=
Z
Z
1
1 + 2 cos 2x + cos2 2x dx
4
1
1
1 + 2 cos 2x + [1 + cos 4x]
4
2
=
x sin 2x x sin 4x
+
+ +
+C
4
4
8
32
=
3x sin 2x sin 4x
+
+
+ C.
8
4
32
5
dx
dx.
12.7.3 EXERCISES
1. Determine the indefinite integral
Z
cos x cos 3x dx.
2. Evaluate the definite integral
Z
π
3
cos 4x sin 2x dx.
0
3. Determine the following indefinite integrals:
(a)
Z
sin5 x dx;
Z
cos3 x dx.
(b)
4. Evaluate the following definite integrals:
(a)
π
8
Z
sin4 x dx;
0
(b)
π
2
Z
cos6 x dx.
0
12.7.4 ANSWERS TO EXERCISES
1.
sin 4x sin 2x
+
+ C.
8
4
2.
√
−
3
' −0.433
4
6
3. (a)
−
cos5 x
cos3 x
+2
− cos x + C,
5
3
or
1
cos 5x 5 cos 3x
−
+
− 10 cos x + C by complex numbers;
16
5
3
(b)
sin x −
sin3 x
+ C,
3
or
1 sin 3x
− 3 sin x + C by complex numbers.
4
3
4. (a)
1.735 × 10−3 approx;
(b)
−1.
7