let A be non empty closed_interval Subset of REAL; for f, f1 being PartFunc of REAL,REAL
for Z being open Subset of REAL st A c= Z & f1 = #Z 2 & f = (- ((sin / cos) / f1)) + (((id Z) ^) / (cos ^2)) & Z c= dom (((id Z) ^) (#) tan) & Z = dom f & f | A is continuous holds
integral (f,A) = ((((id Z) ^) (#) tan) . (upper_bound A)) - ((((id Z) ^) (#) tan) . (lower_bound A))
let f, f1 be PartFunc of REAL,REAL; for Z being open Subset of REAL st A c= Z & f1 = #Z 2 & f = (- ((sin / cos) / f1)) + (((id Z) ^) / (cos ^2)) & Z c= dom (((id Z) ^) (#) tan) & Z = dom f & f | A is continuous holds
integral (f,A) = ((((id Z) ^) (#) tan) . (upper_bound A)) - ((((id Z) ^) (#) tan) . (lower_bound A))
let Z be open Subset of REAL; ( A c= Z & f1 = #Z 2 & f = (- ((sin / cos) / f1)) + (((id Z) ^) / (cos ^2)) & Z c= dom (((id Z) ^) (#) tan) & Z = dom f & f | A is continuous implies integral (f,A) = ((((id Z) ^) (#) tan) . (upper_bound A)) - ((((id Z) ^) (#) tan) . (lower_bound A)) )
assume A1:
( A c= Z & f1 = #Z 2 & f = (- ((sin / cos) / f1)) + (((id Z) ^) / (cos ^2)) & Z c= dom (((id Z) ^) (#) tan) & Z = dom f & f | A is continuous )
; integral (f,A) = ((((id Z) ^) (#) tan) . (upper_bound A)) - ((((id Z) ^) (#) tan) . (lower_bound A))
then A2:
( f is_integrable_on A & f | A is bounded )
by INTEGRA5:10, INTEGRA5:11;
set g = id Z;
Z c= (dom ((id Z) ^)) /\ (dom tan)
by A1, VALUED_1:def 4;
then A3:
Z c= dom ((id Z) ^)
by XBOOLE_1:18;
A4:
not 0 in Z
then A6:
((id Z) ^) (#) tan is_differentiable_on Z
by A1, FDIFF_8:34;
dom f = (dom (- ((sin / cos) / f1))) /\ (dom (((id Z) ^) / (cos ^2)))
by A1, VALUED_1:def 1;
then
( dom f c= dom (- ((sin / cos) / f1)) & dom f c= dom (((id Z) ^) / (cos ^2)) )
by XBOOLE_1:18;
then A7:
( Z c= dom ((sin / cos) / f1) & Z c= dom (((id Z) ^) / (cos ^2)) )
by A1, VALUED_1:8;
dom ((sin / cos) / f1) = (dom (sin / cos)) /\ ((dom f1) \ (f1 " {0}))
by RFUNCT_1:def 1;
then A8:
Z c= dom (sin / cos)
by A7, XBOOLE_1:18;
dom (((id Z) ^) / (cos ^2)) c= (dom ((id Z) ^)) /\ ((dom (cos ^2)) \ ((cos ^2) " {0}))
by RFUNCT_1:def 1;
then
( dom (((id Z) ^) / (cos ^2)) c= dom ((id Z) ^) & dom (((id Z) ^) / (cos ^2)) c= (dom (cos ^2)) \ ((cos ^2) " {0}) )
by XBOOLE_1:18;
then A9:
( Z c= dom ((id Z) ^) & Z c= (dom (cos ^2)) \ ((cos ^2) " {0}) )
by A7;
A10:
for x being Real st x in Z holds
f . x = (- (((sin . x) / (cos . x)) / (x ^2))) + ((1 / x) / ((cos . x) ^2))
proof
let x be
Real;
( x in Z implies f . x = (- (((sin . x) / (cos . x)) / (x ^2))) + ((1 / x) / ((cos . x) ^2)) )
assume A11:
x in Z
;
f . x = (- (((sin . x) / (cos . x)) / (x ^2))) + ((1 / x) / ((cos . x) ^2))
then ((- ((sin / cos) / f1)) + (((id Z) ^) / (cos ^2))) . x =
((- ((sin / cos) / f1)) . x) + ((((id Z) ^) / (cos ^2)) . x)
by A1, VALUED_1:def 1
.=
(- (((sin / cos) / f1) . x)) + ((((id Z) ^) / (cos ^2)) . x)
by VALUED_1:8
.=
(- (((sin / cos) . x) / (f1 . x))) + ((((id Z) ^) / (cos ^2)) . x)
by A11, A7, RFUNCT_1:def 1
.=
(- (((sin . x) * ((cos . x) ")) / (f1 . x))) + ((((id Z) ^) / (cos ^2)) . x)
by A8, A11, RFUNCT_1:def 1
.=
(- (((sin . x) / (cos . x)) / (f1 . x))) + ((((id Z) ^) . x) / ((cos ^2) . x))
by A7, A11, RFUNCT_1:def 1
.=
(- (((sin . x) / (cos . x)) / (f1 . x))) + ((((id Z) . x) ") / ((cos ^2) . x))
by A9, A11, RFUNCT_1:def 2
.=
(- (((sin . x) / (cos . x)) / (f1 . x))) + ((1 / x) / ((cos ^2) . x))
by A11, FUNCT_1:18
.=
(- (((sin . x) / (cos . x)) / (f1 . x))) + ((1 / x) / ((cos . x) ^2))
by VALUED_1:11
.=
(- (((sin . x) / (cos . x)) / (x #Z 2))) + ((1 / x) / ((cos . x) ^2))
by A1, TAYLOR_1:def 1
.=
(- (((sin . x) / (cos . x)) / (x ^2))) + ((1 / x) / ((cos . x) ^2))
by FDIFF_7:1
;
hence
f . x = (- (((sin . x) / (cos . x)) / (x ^2))) + ((1 / x) / ((cos . x) ^2))
by A1;
verum
end;
A12:
for x being Element of REAL st x in dom ((((id Z) ^) (#) tan) `| Z) holds
((((id Z) ^) (#) tan) `| Z) . x = f . x
dom ((((id Z) ^) (#) tan) `| Z) = dom f
by A1, A6, FDIFF_1:def 7;
then
(((id Z) ^) (#) tan) `| Z = f
by A12, PARTFUN1:5;
hence
integral (f,A) = ((((id Z) ^) (#) tan) . (upper_bound A)) - ((((id Z) ^) (#) tan) . (lower_bound A))
by A1, A2, A6, INTEGRA5:13; verum