let A be non empty closed_interval Subset of REAL; for f being PartFunc of REAL,REAL
for Z being open Subset of REAL st A c= Z & Z = dom f & f = (exp_R * cos) (#) sin holds
integral (f,A) = ((- (exp_R * cos)) . (upper_bound A)) - ((- (exp_R * cos)) . (lower_bound A))
let f be PartFunc of REAL,REAL; for Z being open Subset of REAL st A c= Z & Z = dom f & f = (exp_R * cos) (#) sin holds
integral (f,A) = ((- (exp_R * cos)) . (upper_bound A)) - ((- (exp_R * cos)) . (lower_bound A))
let Z be open Subset of REAL; ( A c= Z & Z = dom f & f = (exp_R * cos) (#) sin implies integral (f,A) = ((- (exp_R * cos)) . (upper_bound A)) - ((- (exp_R * cos)) . (lower_bound A)) )
assume A1:
( A c= Z & Z = dom f & f = (exp_R * cos) (#) sin )
; integral (f,A) = ((- (exp_R * cos)) . (upper_bound A)) - ((- (exp_R * cos)) . (lower_bound A))
then
Z = (dom (exp_R * cos)) /\ (dom sin)
by VALUED_1:def 4;
then A2:
Z c= dom (exp_R * cos)
by XBOOLE_1:18;
then A3:
exp_R * cos is_differentiable_on Z
by FDIFF_7:36;
sin is_differentiable_on Z
by FDIFF_1:26, SIN_COS:68;
then
f | Z is continuous
by A1, A3, FDIFF_1:21, FDIFF_1:25;
then
f | A is continuous
by A1, FCONT_1:16;
then A4:
( f is_integrable_on A & f | A is bounded )
by A1, INTEGRA5:10, INTEGRA5:11;
A5:
Z c= dom (- (exp_R * cos))
by A2, VALUED_1:8;
then A6:
(- 1) (#) (exp_R * cos) is_differentiable_on Z
by A3, FDIFF_1:20;
A7:
for x being Real st x in Z holds
((- (exp_R * cos)) `| Z) . x = (exp_R . (cos . x)) * (sin . x)
proof
let x be
Real;
( x in Z implies ((- (exp_R * cos)) `| Z) . x = (exp_R . (cos . x)) * (sin . x) )
assume A8:
x in Z
;
((- (exp_R * cos)) `| Z) . x = (exp_R . (cos . x)) * (sin . x)
A9:
cos is_differentiable_in x
by SIN_COS:63;
A10:
exp_R is_differentiable_in cos . x
by SIN_COS:65;
A11:
exp_R * cos is_differentiable_in x
by A3, A8, FDIFF_1:9;
((- (exp_R * cos)) `| Z) . x =
diff (
(- (exp_R * cos)),
x)
by A6, A8, FDIFF_1:def 7
.=
(- 1) * (diff ((exp_R * cos),x))
by A11, FDIFF_1:15
.=
(- 1) * ((diff (exp_R,(cos . x))) * (diff (cos,x)))
by A9, A10, FDIFF_2:13
.=
(- 1) * ((diff (exp_R,(cos . x))) * (- (sin . x)))
by SIN_COS:63
.=
(- 1) * ((exp_R . (cos . x)) * (- (sin . x)))
by SIN_COS:65
.=
(exp_R . (cos . x)) * (sin . x)
;
hence
((- (exp_R * cos)) `| Z) . x = (exp_R . (cos . x)) * (sin . x)
;
verum
end;
A12:
for x being Real st x in Z holds
f . x = (exp_R . (cos . x)) * (sin . x)
A14:
for x being Element of REAL st x in dom ((- (exp_R * cos)) `| Z) holds
((- (exp_R * cos)) `| Z) . x = f . x
dom ((- (exp_R * cos)) `| Z) = dom f
by A1, A6, FDIFF_1:def 7;
then
(- (exp_R * cos)) `| Z = f
by A14, PARTFUN1:5;
hence
integral (f,A) = ((- (exp_R * cos)) . (upper_bound A)) - ((- (exp_R * cos)) . (lower_bound A))
by A1, A4, A3, A5, FDIFF_1:20, INTEGRA5:13; verum