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 & ( for x being Real st x in Z holds
f1 . x = 1 ) & f = (arctan / (#Z 2)) - (((id Z) (#) (f1 + (#Z 2))) ^) & Z c= dom (((id Z) ^) (#) arctan) & Z c= ].(- 1),1.[ & Z = dom f & f | A is continuous holds
integral (f,A) = ((- (((id Z) ^) (#) arctan)) . (upper_bound A)) - ((- (((id Z) ^) (#) arctan)) . (lower_bound A))
let f, f1 be PartFunc of REAL,REAL; for Z being open Subset of REAL st A c= Z & ( for x being Real st x in Z holds
f1 . x = 1 ) & f = (arctan / (#Z 2)) - (((id Z) (#) (f1 + (#Z 2))) ^) & Z c= dom (((id Z) ^) (#) arctan) & Z c= ].(- 1),1.[ & Z = dom f & f | A is continuous holds
integral (f,A) = ((- (((id Z) ^) (#) arctan)) . (upper_bound A)) - ((- (((id Z) ^) (#) arctan)) . (lower_bound A))
let Z be open Subset of REAL; ( A c= Z & ( for x being Real st x in Z holds
f1 . x = 1 ) & f = (arctan / (#Z 2)) - (((id Z) (#) (f1 + (#Z 2))) ^) & Z c= dom (((id Z) ^) (#) arctan) & Z c= ].(- 1),1.[ & Z = dom f & f | A is continuous implies integral (f,A) = ((- (((id Z) ^) (#) arctan)) . (upper_bound A)) - ((- (((id Z) ^) (#) arctan)) . (lower_bound A)) )
assume A1:
( A c= Z & ( for x being Real st x in Z holds
f1 . x = 1 ) & f = (arctan / (#Z 2)) - (((id Z) (#) (f1 + (#Z 2))) ^) & Z c= dom (((id Z) ^) (#) arctan) & Z c= ].(- 1),1.[ & Z = dom f & f | A is continuous )
; integral (f,A) = ((- (((id Z) ^) (#) arctan)) . (upper_bound A)) - ((- (((id Z) ^) (#) arctan)) . (lower_bound A))
then A2:
( f is_integrable_on A & f | A is bounded )
by INTEGRA5:10, INTEGRA5:11;
A3:
- (((id Z) ^) (#) arctan) is_differentiable_on Z
by A1, Th54;
A4:
Z = (dom (arctan / (#Z 2))) /\ (dom (((id Z) (#) (f1 + (#Z 2))) ^))
by A1, VALUED_1:12;
then A5:
Z c= dom (arctan / (#Z 2))
by XBOOLE_1:18;
A6:
Z c= dom (((id Z) (#) (f1 + (#Z 2))) ^)
by A4, XBOOLE_1:18;
dom (((id Z) (#) (f1 + (#Z 2))) ^) c= dom ((id Z) (#) (f1 + (#Z 2)))
by RFUNCT_1:1;
then
Z c= dom ((id Z) (#) (f1 + (#Z 2)))
by A6;
then
Z c= (dom (id Z)) /\ (dom (f1 + (#Z 2)))
by VALUED_1:def 4;
then A7:
Z c= dom (f1 + (#Z 2))
by XBOOLE_1:18;
A8:
for x being Real st x in Z holds
f . x = ((arctan . x) / (x ^2)) - (1 / (x * (1 + (x ^2))))
A11:
for x being Element of REAL st x in dom ((- (((id Z) ^) (#) arctan)) `| Z) holds
((- (((id Z) ^) (#) arctan)) `| Z) . x = f . x
dom ((- (((id Z) ^) (#) arctan)) `| Z) = dom f
by A1, A3, FDIFF_1:def 7;
then
(- (((id Z) ^) (#) arctan)) `| Z = f
by A11, PARTFUN1:5;
hence
integral (f,A) = ((- (((id Z) ^) (#) arctan)) . (upper_bound A)) - ((- (((id Z) ^) (#) arctan)) . (lower_bound A))
by A1, A2, A3, INTEGRA5:13; verum