1 Question 409: Integrals and Fractals Edgar Valdebenito abstract This note presents some definite integrals. 1. Intoduction. Some definite integrals. 2 1 1 0 2 5 5 25 3 cos 2 4 25 x dx (1) 2 1 1 0 5 25 2 5 6 sin 4 25 x dx (2) 2 2 1 1 2 0 2 1 4 cos 3 1 x z z dx z (3) 2 2 1 1 2 0 1 2 4 sin 1 x z z dx z (4) In (3) , (4) : 1/3 1/3 11 2 68 199 3 33 3 3 3 199 3 33 z (5) 2 2 1 1 2 0 2 1 7 cos 5 1 x z z dx z (6) 2 2 1 1 2 0 1 2 14 sin 3 1 x z z dx z (7) In (6) , (7) :
Transcript
1
Question 409: Integrals and Fractals
Edgar Valdebenito
abstract
This note presents some definite integrals.
1. Intoduction. Some definite integrals.
21
1
0
2 5 5 2 53cos
2 4 2 5
x
dx
(1)
21
1
0
5 2 5 2 56 sin
4 2 5
x
dx
(2)
2 21
1
2
0
2 14cos
3 1
xz zdx
z
(3)
2 21
1
2
0
1 24 sin
1
xz zdx
z
(4)
In (3) , (4) :
1/3
1/3
11 2 68199 3 33
3 3 3 199 3 33
z
(5)
2 21
1
2
0
2 17cos
5 1
xz zdx
z
(6)
2 21
1
2
0
1 214sin
3 1
xz zdx
z
(7)
In (6) , (7) :
2
1/3
1/3
7 1 52388 12 69
3 3 3 388 12 69
z
(8)
2 21
1
2
0
2 16cos
5 1
xz zdx
z
(9)
2 21
1
2
0
1 23 sin
1
xz zdx
z
(10)
In (9) , (10) : 57.7341...z is root of the equation:
5 4 3 257 42 22 7 1 0z z z z z (11)
2. The equation 5 4 3 257 42 22 7 1 0z z z z z .
The equation
5 4 3 257 42 22 7 1 0f x x x x x x (12)
Is not solvable by radicals. Galois group G f is not soluble.
1
2
3
4
5
57.7341095724734413...
0.0925... 0.4268...
0 0.0925... 0.4268...
0.2745... 0.1242...
0.2745... 0.1242...
x
x i
f x x i
x i
x i
(13)
3. Relations
5 4 3 257 42 22 7 1f x x x x x x (14)
6 51 64g x x x (15)
1g x x f x (16)
4. Representations for root 1 57.7341...x z
3
66
1 1 1 1 1...
2 2 2 2 2y
(17)
6
1x z y (18)
5. Iterative methods
5
1 1 164 1, 57 57.73410...1
nn n
n
uu u u x z
u
(19)
6
1 1
1
1 1 1, 0 0.01732...
2
nn n
vv v v
x z
(20)
Figure 1.
Figure 2.
4
6. Fractals
Fractals for 5
64 11
xF x x
x
Figure 3.
5
Figure 4.
6
Figure 5.
7
Figure 6.
8
Figure 7.
9
Figure 8.
10
Figure 9.
11
Figure 10.
12
Figure 11.
13
Figure 12.
14
Fractals for 5 4 3 257 42 22 7 1F x x x x x x .
Figure 13.
15
Figure 14.
16
Figure 15.
17
Figure 16.
18
References 1. Boros, G. and Moll, V.H.: Irresistible Integrals, Cambridge University Press, 2004.
2. Falconer, K.: Fractal Geometry : Mathematical Foundations and Applications. John Wiley &
Sons, Ltd.,2003, pp.XXV. ISBN-0-470-84862-6.
3. Jacquin, A.E.: Image coding based on a fractal theory of iterated contractive image
