let A be non empty closed_interval Subset of REAL; for f, f1, f2 being PartFunc of REAL,REAL
for Z being open Subset of REAL st A c= Z & Z c= dom (ln * (f1 + f2)) & f = (id Z) / (f1 + f2) & f2 = #Z 2 & ( for x being Real st x in Z holds
f1 . x = 1 ) & Z = dom f & f | A is continuous holds
integral (f,A) = (((1 / 2) (#) (ln * (f1 + f2))) . (upper_bound A)) - (((1 / 2) (#) (ln * (f1 + f2))) . (lower_bound A))
let f, f1, f2 be PartFunc of REAL,REAL; for Z being open Subset of REAL st A c= Z & Z c= dom (ln * (f1 + f2)) & f = (id Z) / (f1 + f2) & f2 = #Z 2 & ( for x being Real st x in Z holds
f1 . x = 1 ) & Z = dom f & f | A is continuous holds
integral (f,A) = (((1 / 2) (#) (ln * (f1 + f2))) . (upper_bound A)) - (((1 / 2) (#) (ln * (f1 + f2))) . (lower_bound A))
let Z be open Subset of REAL; ( A c= Z & Z c= dom (ln * (f1 + f2)) & f = (id Z) / (f1 + f2) & f2 = #Z 2 & ( for x being Real st x in Z holds
f1 . x = 1 ) & Z = dom f & f | A is continuous implies integral (f,A) = (((1 / 2) (#) (ln * (f1 + f2))) . (upper_bound A)) - (((1 / 2) (#) (ln * (f1 + f2))) . (lower_bound A)) )
assume A1:
( A c= Z & Z c= dom (ln * (f1 + f2)) & f = (id Z) / (f1 + f2) & f2 = #Z 2 & ( for x being Real st x in Z holds
f1 . x = 1 ) & Z = dom f & f | A is continuous )
; integral (f,A) = (((1 / 2) (#) (ln * (f1 + f2))) . (upper_bound A)) - (((1 / 2) (#) (ln * (f1 + f2))) . (lower_bound A))
then A2:
( f is_integrable_on A & f | A is bounded )
by INTEGRA5:10, INTEGRA5:11;
A3:
Z c= dom ((1 / 2) (#) (ln * (f1 + f2)))
by A1, VALUED_1:def 5;
Z c= (dom (id Z)) /\ ((dom (f1 + f2)) \ ((f1 + f2) " {0}))
by A1, RFUNCT_1:def 1;
then
Z c= (dom (f1 + f2)) \ ((f1 + f2) " {0})
by XBOOLE_1:18;
then A4:
Z c= dom ((f1 + f2) ^)
by RFUNCT_1:def 2;
dom ((f1 + f2) ^) c= dom (f1 + f2)
by RFUNCT_1:1;
then A5:
Z c= dom (f1 + f2)
by A4;
A6:
(1 / 2) (#) (ln * (f1 + f2)) is_differentiable_on Z
by A1, A3, SIN_COS9:102;
A7:
for x being Real st x in Z holds
f . x = x / (1 + (x ^2))
A9:
for x being Element of REAL st x in dom (((1 / 2) (#) (ln * (f1 + f2))) `| Z) holds
(((1 / 2) (#) (ln * (f1 + f2))) `| Z) . x = f . x
dom (((1 / 2) (#) (ln * (f1 + f2))) `| Z) = dom f
by A1, A6, FDIFF_1:def 7;
then
((1 / 2) (#) (ln * (f1 + f2))) `| Z = f
by A9, PARTFUN1:5;
hence
integral (f,A) = (((1 / 2) (#) (ln * (f1 + f2))) . (upper_bound A)) - (((1 / 2) (#) (ln * (f1 + f2))) . (lower_bound A))
by A1, A2, A6, INTEGRA5:13; verum