Article INTEGRA8, MML version 4.99.1005
:: INTEGRA8:th 1
theorem
for b1 being Element of REAL
for b2 being Element of NAT holds
sin (b1 + ((2 * b2) * PI)) = sin b1;
:: INTEGRA8:th 2
theorem
for b1 being Element of REAL
for b2 being Element of NAT holds
sin (b1 + (((2 * b2) + 1) * PI)) = - sin b1;
:: INTEGRA8:th 3
theorem
for b1 being Element of REAL
for b2 being Element of NAT holds
cos (b1 + ((2 * b2) * PI)) = cos b1;
:: INTEGRA8:th 4
theorem
for b1 being Element of REAL
for b2 being Element of NAT holds
cos (b1 + (((2 * b2) + 1) * PI)) = - cos b1;
:: INTEGRA8:th 5
theorem
for b1 being Element of REAL
st 0 <= sin (b1 / 2)
holds sin (b1 / 2) = sqrt ((1 - cos b1) / 2);
:: INTEGRA8:th 6
theorem
for b1 being Element of REAL
st sin (b1 / 2) < 0
holds sin (b1 / 2) = - sqrt ((1 - cos b1) / 2);
:: INTEGRA8:th 7
theorem
sin (PI / 4) = (sqrt 2) / 2;
:: INTEGRA8:th 8
theorem
sin - (PI / 4) = - ((sqrt 2) / 2);
:: INTEGRA8:th 9
theorem
[.- ((sqrt 2) / 2),(sqrt 2) / 2.] c= ].- 1,1.[;
:: INTEGRA8:th 10
theorem
arcsin ((sqrt 2) / 2) = PI / 4;
:: INTEGRA8:th 11
theorem
arcsin - ((sqrt 2) / 2) = - (PI / 4);
:: INTEGRA8:th 12
theorem
for b1 being Element of REAL
st 0 <= cos (b1 / 2)
holds cos (b1 / 2) = sqrt ((1 + cos b1) / 2);
:: INTEGRA8:th 13
theorem
cos (PI / 4) = (sqrt 2) / 2;
:: INTEGRA8:th 14
theorem
cos ((3 * PI) / 4) = - ((sqrt 2) / 2);
:: INTEGRA8:th 15
theorem
arccos ((sqrt 2) / 2) = PI / 4;
:: INTEGRA8:th 16
theorem
arccos - ((sqrt 2) / 2) = (3 * PI) / 4;
:: INTEGRA8:th 17
theorem
sinh . 1 = (number_e ^2 - 1) / (2 * number_e);
:: INTEGRA8:th 18
theorem
cosh . 0 = 1;
:: INTEGRA8:th 19
theorem
cosh . 1 = (number_e ^2 + 1) / (2 * number_e);
:: INTEGRA8:th 20
theorem
for b1 being Function-like linear Relation of REAL,REAL holds
- b1 is Function-like linear Relation of REAL,REAL;
:: INTEGRA8:th 21
theorem
for b1 being Function-like REST-like Relation of REAL,REAL holds
- b1 is Function-like REST-like Relation of REAL,REAL;
:: INTEGRA8:th 22
theorem
for b1 being Function-like Relation of REAL,REAL
for b2 being Element of REAL
st b1 is_differentiable_in b2
holds - b1 is_differentiable_in b2 & diff(- b1,b2) = - diff(b1,b2);
:: INTEGRA8:th 23
theorem
for b1 being Function-like Relation of REAL,REAL
for b2 being open Element of bool REAL
st b2 c= proj1 - b1 & b1 is_differentiable_on b2
holds - b1 is_differentiable_on b2 &
(for b3 being Element of REAL
st b3 in b2
holds ((- b1) `| b2) . b3 = - diff(b1,b3));
:: INTEGRA8:th 24
theorem
- sin is_differentiable_on REAL;
:: INTEGRA8:th 25
theorem
for b1 being Element of REAL holds
- cos is_differentiable_in b1 &
diff(- cos,b1) = sin . b1;
:: INTEGRA8:th 26
theorem
- cos is_differentiable_on REAL &
(for b1 being Element of REAL
st b1 in REAL
holds diff(- cos,b1) = sin . b1);
:: INTEGRA8:th 27
theorem
sin `| REAL = cos;
:: INTEGRA8:th 28
theorem
cos `| REAL = - sin;
:: INTEGRA8:th 29
theorem
(- cos) `| REAL = sin;
:: INTEGRA8:th 30
theorem
sinh `| REAL = cosh;
:: INTEGRA8:th 31
theorem
cosh `| REAL = sinh;
:: INTEGRA8:th 32
theorem
exp_R `| REAL = exp_R;
:: INTEGRA8:th 33
theorem
for b1 being Function-like Relation of REAL,REAL
for b2 being open Element of bool REAL
st b2 c= proj1 tan &
(for b3 being Element of REAL
st b3 in b2
holds b1 . b3 = 1 / ((cos . b3) ^2) &
cos . b3 <> 0)
holds tan is_differentiable_on b2 &
(for b3 being Element of REAL
st b3 in b2
holds (tan `| b2) . b3 = 1 / ((cos . b3) ^2));
:: INTEGRA8:th 34
theorem
for b1 being Function-like Relation of REAL,REAL
for b2 being open Element of bool REAL
st b2 c= proj1 cot &
(for b3 being Element of REAL
st b3 in b2
holds b1 . b3 = - (1 / ((sin . b3) ^2)) &
sin . b3 <> 0)
holds cot is_differentiable_on b2 &
(for b3 being Element of REAL
st b3 in b2
holds (cot `| b2) . b3 = - (1 / ((sin . b3) ^2)));
:: INTEGRA8:th 35
theorem
for b1 being Element of REAL holds
proj1 (REAL --> b1) = REAL & proj2 (REAL --> b1) c= REAL;
:: INTEGRA8:funcnot 1 => INTEGRA8:func 1
definition
let a1 be Element of REAL;
func Cst A1 -> Function-like quasi_total Relation of REAL,REAL equals
REAL --> a1;
end;
:: INTEGRA8:def 1
theorem
for b1 being Element of REAL holds
Cst b1 = REAL --> b1;
:: INTEGRA8:th 36
theorem
for b1, b2 being Element of REAL
for b3 being closed-interval Element of bool REAL holds
chi(b3,b3) = (Cst 1) | b3;
:: INTEGRA8:th 37
theorem
for b1, b2 being Element of REAL
for b3 being closed-interval Element of bool REAL
st b3 = [.b1,b2.]
holds sup b3 = b2 & inf b3 = b1;
:: INTEGRA8:th 38
theorem
for b1, b2 being Element of REAL
st b1 <= b2
holds integral(Cst 1,b1,b2) = b2 - b1;
:: INTEGRA8:th 39
theorem
for b1 being closed-interval Element of bool REAL holds
integral(cos,b1) = (sin . sup b1) - (sin . inf b1);
:: INTEGRA8:th 40
theorem
for b1 being closed-interval Element of bool REAL
st b1 = [.0,PI / 2.]
holds integral(cos,b1) = 1;
:: INTEGRA8:th 41
theorem
for b1 being closed-interval Element of bool REAL
st b1 = [.0,PI.]
holds integral(cos,b1) = 0;
:: INTEGRA8:th 42
theorem
for b1 being closed-interval Element of bool REAL
st b1 = [.0,(PI * 3) / 2.]
holds integral(cos,b1) = - 1;
:: INTEGRA8:th 43
theorem
for b1 being closed-interval Element of bool REAL
st b1 = [.0,PI * 2.]
holds integral(cos,b1) = 0;
:: INTEGRA8:th 44
theorem
for b1 being closed-interval Element of bool REAL
for b2 being Element of NAT
st b1 = [.(2 * b2) * PI,((2 * b2) + 1) * PI.]
holds integral(cos,b1) = 0;
:: INTEGRA8:th 45
theorem
for b1 being closed-interval Element of bool REAL
for b2 being Element of REAL
for b3 being Element of NAT
st b1 = [.b2 + ((2 * b3) * PI),b2 + (((2 * b3) + 1) * PI).]
holds integral(cos,b1) = - (2 * sin b2);
:: INTEGRA8:th 46
theorem
for b1 being closed-interval Element of bool REAL holds
integral(- sin,b1) = (cos . sup b1) - (cos . inf b1);
:: INTEGRA8:th 47
theorem
for b1 being closed-interval Element of bool REAL
st b1 = [.0,PI / 2.]
holds integral(- sin,b1) = - 1;
:: INTEGRA8:th 48
theorem
for b1 being closed-interval Element of bool REAL
st b1 = [.0,PI.]
holds integral(- sin,b1) = - 2;
:: INTEGRA8:th 49
theorem
for b1 being closed-interval Element of bool REAL
st b1 = [.0,(PI * 3) / 2.]
holds integral(- sin,b1) = - 1;
:: INTEGRA8:th 50
theorem
for b1 being closed-interval Element of bool REAL
st b1 = [.0,PI * 2.]
holds integral(- sin,b1) = 0;
:: INTEGRA8:th 51
theorem
for b1 being closed-interval Element of bool REAL
for b2 being Element of NAT
st b1 = [.(2 * b2) * PI,((2 * b2) + 1) * PI.]
holds integral(- sin,b1) = - 2;
:: INTEGRA8:th 52
theorem
for b1 being closed-interval Element of bool REAL
for b2 being Element of REAL
for b3 being Element of NAT
st b1 = [.b2 + ((2 * b3) * PI),b2 + (((2 * b3) + 1) * PI).]
holds integral(- sin,b1) = - (2 * cos b2);
:: INTEGRA8:th 53
theorem
for b1 being closed-interval Element of bool REAL holds
integral(exp_R,b1) = (exp_R . sup b1) - (exp_R . inf b1);
:: INTEGRA8:th 54
theorem
for b1 being closed-interval Element of bool REAL
st b1 = [.0,1.]
holds integral(exp_R,b1) = number_e - 1;
:: INTEGRA8:th 55
theorem
for b1 being closed-interval Element of bool REAL holds
integral(sinh,b1) = (cosh . sup b1) - (cosh . inf b1);
:: INTEGRA8:th 56
theorem
for b1 being closed-interval Element of bool REAL
st b1 = [.0,1.]
holds integral(sinh,b1) = (number_e - 1) ^2 / (2 * number_e);
:: INTEGRA8:th 57
theorem
for b1 being closed-interval Element of bool REAL holds
integral(cosh,b1) = (sinh . sup b1) - (sinh . inf b1);
:: INTEGRA8:th 58
theorem
for b1 being closed-interval Element of bool REAL
st b1 = [.0,1.]
holds integral(cosh,b1) = (number_e ^2 - 1) / (2 * number_e);
:: INTEGRA8:th 59
theorem
for b1 being Function-like Relation of REAL,REAL
for b2 being closed-interval Element of bool REAL
for b3 being open Element of bool REAL
st b2 c= b3 &
proj1 tan = b3 &
proj1 tan = proj1 b1 &
(for b4 being Element of REAL
st b4 in b3
holds b1 . b4 = 1 / ((cos . b4) ^2) &
cos . b4 <> 0) &
b1 is_continuous_on b2
holds integral(b1,b2) = (tan . sup b2) - (tan . inf b2);
:: INTEGRA8:th 60
theorem
for b1 being Function-like Relation of REAL,REAL
for b2 being closed-interval Element of bool REAL
for b3 being open Element of bool REAL
st b2 c= b3 &
proj1 cot = b3 &
proj1 cot = proj1 b1 &
(for b4 being Element of REAL
st b4 in b3
holds b1 . b4 = - (1 / ((sin . b4) ^2)) &
sin . b4 <> 0) &
b1 is_continuous_on b2
holds integral(b1,b2) = (cot . sup b2) - (cot . inf b2);
:: INTEGRA8:th 61
theorem
for b1 being Function-like Relation of REAL,REAL
for b2 being closed-interval Element of bool REAL
st proj1 tanh = proj1 b1 &
(for b3 being Element of REAL
st b3 in REAL
holds b1 . b3 = 1 / ((cosh . b3) ^2)) &
b1 is_continuous_on b2
holds integral(b1,b2) = (tanh . sup b2) - (tanh . inf b2);
:: INTEGRA8:th 62
theorem
for b1 being Function-like Relation of REAL,REAL
for b2 being closed-interval Element of bool REAL
st b2 c= ].- 1,1.[ &
proj1 (arcsin `| ].- 1,1.[) = proj1 b1 &
(for b3 being Element of REAL holds
b3 in ].- 1,1.[ &
b1 . b3 = 1 / sqrt (1 - (b3 ^2))) &
b1 is_continuous_on b2
holds integral(b1,b2) = (arcsin . sup b2) - (arcsin . inf b2);
:: INTEGRA8:th 63
theorem
for b1 being Function-like Relation of REAL,REAL
for b2 being closed-interval Element of bool REAL
st b2 c= ].- 1,1.[ &
proj1 (arccos `| ].- 1,1.[) = proj1 b1 &
(for b3 being Element of REAL holds
b3 in ].- 1,1.[ &
b1 . b3 = - (1 / sqrt (1 - (b3 ^2)))) &
b1 is_continuous_on b2
holds integral(b1,b2) = (arccos . sup b2) - (arccos . inf b2);
:: INTEGRA8:th 64
theorem
for b1 being Function-like Relation of REAL,REAL
for b2 being closed-interval Element of bool REAL
st b2 = [.- ((sqrt 2) / 2),(sqrt 2) / 2.] &
proj1 (arcsin `| ].- 1,1.[) = proj1 b1 &
(for b3 being Element of REAL holds
b3 in ].- 1,1.[ &
b1 . b3 = 1 / sqrt (1 - (b3 ^2))) &
b1 is_continuous_on b2
holds integral(b1,b2) = PI / 2;
:: INTEGRA8:th 65
theorem
for b1 being Function-like Relation of REAL,REAL
for b2 being closed-interval Element of bool REAL
st b2 = [.- ((sqrt 2) / 2),(sqrt 2) / 2.] &
proj1 (arccos `| ].- 1,1.[) = proj1 b1 &
(for b3 being Element of REAL holds
b3 in ].- 1,1.[ &
b1 . b3 = - (1 / sqrt (1 - (b3 ^2)))) &
b1 is_continuous_on b2
holds integral(b1,b2) = - (PI / 2);
:: INTEGRA8:th 66
theorem
for b1, b2 being Function-like Relation of REAL,REAL
for b3 being closed-interval Element of bool REAL
for b4 being open Element of bool REAL
st b1 is_differentiable_on b4 & b2 is_differentiable_on b4 & b3 c= b4 & b1 `| b4 is_integrable_on b3 & b1 `| b4 is_bounded_on b3 & b2 `| b4 is_integrable_on b3 & b2 `| b4 is_bounded_on b3
holds integral((b1 `| b4) + (b2 `| b4),b3) = (((b1 . sup b3) - (b1 . inf b3)) + (b2 . sup b3)) - (b2 . inf b3);
:: INTEGRA8:th 67
theorem
for b1, b2 being Function-like Relation of REAL,REAL
for b3 being closed-interval Element of bool REAL
for b4 being open Element of bool REAL
st b1 is_differentiable_on b4 & b2 is_differentiable_on b4 & b3 c= b4 & b1 `| b4 is_integrable_on b3 & b1 `| b4 is_bounded_on b3 & b2 `| b4 is_integrable_on b3 & b2 `| b4 is_bounded_on b3
holds integral((b1 `| b4) - (b2 `| b4),b3) = ((b1 . sup b3) - (b1 . inf b3)) - ((b2 . sup b3) - (b2 . inf b3));
:: INTEGRA8:th 68
theorem
for b1 being Function-like Relation of REAL,REAL
for b2 being closed-interval Element of bool REAL
for b3 being Element of REAL
for b4 being open Element of bool REAL
st b1 is_differentiable_on b4 & b2 c= b4 & b1 `| b4 is_integrable_on b2 & b1 `| b4 is_bounded_on b2
holds integral(b3 (#) (b1 `| b4),b2) = (b3 * (b1 . sup b2)) - (b3 * (b1 . inf b2));
:: INTEGRA8:th 69
theorem
for b1 being closed-interval Element of bool REAL holds
integral(sin + cos,b1) = ((((- cos) . sup b1) - ((- cos) . inf b1)) + (sin . sup b1)) - (sin . inf b1);
:: INTEGRA8:th 70
theorem
for b1 being closed-interval Element of bool REAL
st b1 = [.0,PI / 2.]
holds integral(sin + cos,b1) = 2;
:: INTEGRA8:th 71
theorem
for b1 being closed-interval Element of bool REAL
st b1 = [.0,PI.]
holds integral(sin + cos,b1) = 2;
:: INTEGRA8:th 72
theorem
for b1 being closed-interval Element of bool REAL
st b1 = [.0,(PI * 3) / 2.]
holds integral(sin + cos,b1) = 0;
:: INTEGRA8:th 73
theorem
for b1 being closed-interval Element of bool REAL
st b1 = [.0,PI * 2.]
holds integral(sin + cos,b1) = 0;
:: INTEGRA8:th 74
theorem
for b1 being closed-interval Element of bool REAL
for b2 being Element of NAT
st b1 = [.(2 * b2) * PI,((2 * b2) + 1) * PI.]
holds integral(sin + cos,b1) = 2;
:: INTEGRA8:th 75
theorem
for b1 being closed-interval Element of bool REAL
for b2 being Element of REAL
for b3 being Element of NAT
st b1 = [.b2 + ((2 * b3) * PI),b2 + (((2 * b3) + 1) * PI).]
holds integral(sin + cos,b1) = (2 * cos b2) - (2 * sin b2);
:: INTEGRA8:th 76
theorem
for b1 being closed-interval Element of bool REAL holds
integral(sinh + cosh,b1) = (((cosh . sup b1) - (cosh . inf b1)) + (sinh . sup b1)) - (sinh . inf b1);
:: INTEGRA8:th 77
theorem
for b1 being closed-interval Element of bool REAL
st b1 = [.0,1.]
holds integral(sinh + cosh,b1) = number_e - 1;
:: INTEGRA8:th 78
theorem
for b1 being closed-interval Element of bool REAL holds
integral(sin - cos,b1) = (((- cos) . sup b1) - ((- cos) . inf b1)) - ((sin . sup b1) - (sin . inf b1));
:: INTEGRA8:th 79
theorem
for b1 being closed-interval Element of bool REAL
st b1 = [.0,PI / 2.]
holds integral(sin - cos,b1) = 0;
:: INTEGRA8:th 80
theorem
for b1 being closed-interval Element of bool REAL
st b1 = [.0,PI.]
holds integral(sin - cos,b1) = 2;
:: INTEGRA8:th 81
theorem
for b1 being closed-interval Element of bool REAL
st b1 = [.0,(PI * 3) / 2.]
holds integral(sin - cos,b1) = 2;
:: INTEGRA8:th 82
theorem
for b1 being closed-interval Element of bool REAL
st b1 = [.0,PI * 2.]
holds integral(sin - cos,b1) = 0;
:: INTEGRA8:th 83
theorem
for b1 being closed-interval Element of bool REAL
for b2 being Element of NAT
st b1 = [.(2 * b2) * PI,((2 * b2) + 1) * PI.]
holds integral(sin - cos,b1) = 2;
:: INTEGRA8:th 84
theorem
for b1 being closed-interval Element of bool REAL
for b2 being Element of REAL
for b3 being Element of NAT
st b1 = [.b2 + ((2 * b3) * PI),b2 + (((2 * b3) + 1) * PI).]
holds integral(sin - cos,b1) = (2 * cos b2) + (2 * sin b2);
:: INTEGRA8:th 85
theorem
for b1 being closed-interval Element of bool REAL
for b2 being Element of REAL holds
integral(b2 (#) sin,b1) = (b2 * ((- cos) . sup b1)) - (b2 * ((- cos) . inf b1));
:: INTEGRA8:th 86
theorem
for b1 being closed-interval Element of bool REAL
for b2 being Element of REAL holds
integral(b2 (#) cos,b1) = (b2 * (sin . sup b1)) - (b2 * (sin . inf b1));
:: INTEGRA8:th 87
theorem
for b1 being closed-interval Element of bool REAL
for b2 being Element of REAL holds
integral(b2 (#) sinh,b1) = (b2 * (cosh . sup b1)) - (b2 * (cosh . inf b1));
:: INTEGRA8:th 88
theorem
for b1 being closed-interval Element of bool REAL
for b2 being Element of REAL holds
integral(b2 (#) cosh,b1) = (b2 * (sinh . sup b1)) - (b2 * (sinh . inf b1));
:: INTEGRA8:th 89
theorem
for b1 being closed-interval Element of bool REAL
for b2 being Element of REAL holds
integral(b2 (#) exp_R,b1) = (b2 * (exp_R . sup b1)) - (b2 * (exp_R . inf b1));
:: INTEGRA8:th 90
theorem
for b1 being closed-interval Element of bool REAL holds
integral(sin (#) cos,b1) = (1 / 2) * (((cos . inf b1) * (cos . inf b1)) - ((cos . sup b1) * (cos . sup b1)));
:: INTEGRA8:th 91
theorem
for b1 being closed-interval Element of bool REAL
st b1 = [.0,PI / 2.]
holds integral(sin (#) cos,b1) = 1 / 2;
:: INTEGRA8:th 92
theorem
for b1 being closed-interval Element of bool REAL
st b1 = [.0,PI.]
holds integral(sin (#) cos,b1) = 0;
:: INTEGRA8:th 93
theorem
for b1 being closed-interval Element of bool REAL
st b1 = [.0,PI * (3 / 2).]
holds integral(sin (#) cos,b1) = 1 / 2;
:: INTEGRA8:th 94
theorem
for b1 being closed-interval Element of bool REAL
st b1 = [.0,PI * 2.]
holds integral(sin (#) cos,b1) = 0;
:: INTEGRA8:th 95
theorem
for b1 being closed-interval Element of bool REAL
for b2 being Element of NAT
st b1 = [.(2 * b2) * PI,((2 * b2) + 1) * PI.]
holds integral(sin (#) cos,b1) = 0;
:: INTEGRA8:th 96
theorem
for b1 being closed-interval Element of bool REAL
for b2 being Element of REAL
for b3 being Element of NAT
st b1 = [.b2 + ((2 * b3) * PI),b2 + (((2 * b3) + 1) * PI).]
holds integral(sin (#) cos,b1) = 0;
:: INTEGRA8:th 97
theorem
for b1 being closed-interval Element of bool REAL holds
integral(sin (#) sin,b1) = (((cos . inf b1) * (sin . inf b1)) - ((cos . sup b1) * (sin . sup b1))) + integral(cos (#) cos,b1);
:: INTEGRA8:th 98
theorem
for b1 being closed-interval Element of bool REAL holds
integral(sinh (#) sinh,b1) = (((cosh . sup b1) * (sinh . sup b1)) - ((cosh . inf b1) * (sinh . inf b1))) - integral(cosh (#) cosh,b1);
:: INTEGRA8:th 99
theorem
for b1 being closed-interval Element of bool REAL holds
integral(sinh (#) cosh,b1) = (1 / 2) * (((cosh . sup b1) * (cosh . sup b1)) - ((cosh . inf b1) * (cosh . inf b1)));
:: INTEGRA8:th 100
theorem
for b1 being closed-interval Element of bool REAL holds
integral(exp_R (#) exp_R,b1) = (1 / 2) * ((exp_R . sup b1) ^2 - ((exp_R . inf b1) ^2));
:: INTEGRA8:th 101
theorem
for b1 being closed-interval Element of bool REAL holds
integral(exp_R (#) (sin + cos),b1) = ((exp_R (#) sin) . sup b1) - ((exp_R (#) sin) . inf b1);
:: INTEGRA8:th 102
theorem
for b1 being closed-interval Element of bool REAL holds
integral(exp_R (#) (cos - sin),b1) = ((exp_R (#) cos) . sup b1) - ((exp_R (#) cos) . inf b1);