let r be Real; for A being non empty closed_interval Subset of REAL
for f, g, f1, f2 being PartFunc of REAL,REAL
for Z being open Subset of REAL st A c= Z & Z c= dom (ln * (f1 + f2)) & r <> 0 & ( for x being Real st x in Z holds
( g . x = x / r & g . x > - 1 & g . x < 1 & f1 . x = 1 ) ) & f2 = (#Z 2) * g & Z = dom f & f = arccot * g holds
integral (f,A) = ((((id Z) (#) (arccot * g)) + ((r / 2) (#) (ln * (f1 + f2)))) . (upper_bound A)) - ((((id Z) (#) (arccot * g)) + ((r / 2) (#) (ln * (f1 + f2)))) . (lower_bound A))
let A be non empty closed_interval Subset of REAL; for f, g, f1, f2 being PartFunc of REAL,REAL
for Z being open Subset of REAL st A c= Z & Z c= dom (ln * (f1 + f2)) & r <> 0 & ( for x being Real st x in Z holds
( g . x = x / r & g . x > - 1 & g . x < 1 & f1 . x = 1 ) ) & f2 = (#Z 2) * g & Z = dom f & f = arccot * g holds
integral (f,A) = ((((id Z) (#) (arccot * g)) + ((r / 2) (#) (ln * (f1 + f2)))) . (upper_bound A)) - ((((id Z) (#) (arccot * g)) + ((r / 2) (#) (ln * (f1 + f2)))) . (lower_bound A))
let f, g, f1, f2 be PartFunc of REAL,REAL; for Z being open Subset of REAL st A c= Z & Z c= dom (ln * (f1 + f2)) & r <> 0 & ( for x being Real st x in Z holds
( g . x = x / r & g . x > - 1 & g . x < 1 & f1 . x = 1 ) ) & f2 = (#Z 2) * g & Z = dom f & f = arccot * g holds
integral (f,A) = ((((id Z) (#) (arccot * g)) + ((r / 2) (#) (ln * (f1 + f2)))) . (upper_bound A)) - ((((id Z) (#) (arccot * g)) + ((r / 2) (#) (ln * (f1 + f2)))) . (lower_bound A))
let Z be open Subset of REAL; ( A c= Z & Z c= dom (ln * (f1 + f2)) & r <> 0 & ( for x being Real st x in Z holds
( g . x = x / r & g . x > - 1 & g . x < 1 & f1 . x = 1 ) ) & f2 = (#Z 2) * g & Z = dom f & f = arccot * g implies integral (f,A) = ((((id Z) (#) (arccot * g)) + ((r / 2) (#) (ln * (f1 + f2)))) . (upper_bound A)) - ((((id Z) (#) (arccot * g)) + ((r / 2) (#) (ln * (f1 + f2)))) . (lower_bound A)) )
assume A1:
( A c= Z & Z c= dom (ln * (f1 + f2)) & r <> 0 & ( for x being Real st x in Z holds
( g . x = x / r & g . x > - 1 & g . x < 1 & f1 . x = 1 ) ) & f2 = (#Z 2) * g & Z = dom f & f = arccot * g )
; integral (f,A) = ((((id Z) (#) (arccot * g)) + ((r / 2) (#) (ln * (f1 + f2)))) . (upper_bound A)) - ((((id Z) (#) (arccot * g)) + ((r / 2) (#) (ln * (f1 + f2)))) . (lower_bound A))
Z c= (dom (id Z)) /\ (dom f)
by A1;
then A2:
Z c= dom ((id Z) (#) (arccot * g))
by A1, VALUED_1:def 4;
Z c= dom ((r / 2) (#) (ln * (f1 + f2)))
by A1, VALUED_1:def 5;
then
Z c= (dom ((id Z) (#) (arccot * g))) /\ (dom ((r / 2) (#) (ln * (f1 + f2))))
by A2, XBOOLE_1:19;
then A3:
Z c= dom (((id Z) (#) (arccot * g)) + ((r / 2) (#) (ln * (f1 + f2))))
by VALUED_1:def 1;
A4:
for x being Real st x in Z holds
( g . x = x / r & g . x > - 1 & g . x < 1 )
by A1;
for x being Real st x in Z holds
g . x = ((1 / r) * x) + 0
then
for x being Real st x in Z holds
( g . x = ((1 / r) * x) + 0 & g . x > - 1 & g . x < 1 )
by A1;
then
f is_differentiable_on Z
by A1, SIN_COS9:88;
then
f | Z is continuous
by FDIFF_1:25;
then
f | A is continuous
by A1, FCONT_1:16;
then A5:
( f is_integrable_on A & f | A is bounded )
by A1, INTEGRA5:10, INTEGRA5:11;
A6:
( ( for x being Real st x in Z holds
f1 . x = 1 ) & ( for x being Real st x in Z holds
g . x = x / r ) )
by A1;
then A7:
((id Z) (#) (arccot * g)) + ((r / 2) (#) (ln * (f1 + f2))) is_differentiable_on Z
by A1, A3, A4, SIN_COS9:110;
A8:
for x being Real st x in Z holds
f . x = arccot . (x / r)
A10:
for x being Element of REAL st x in dom ((((id Z) (#) (arccot * g)) + ((r / 2) (#) (ln * (f1 + f2)))) `| Z) holds
((((id Z) (#) (arccot * g)) + ((r / 2) (#) (ln * (f1 + f2)))) `| Z) . x = f . x
dom ((((id Z) (#) (arccot * g)) + ((r / 2) (#) (ln * (f1 + f2)))) `| Z) = dom f
by A1, A7, FDIFF_1:def 7;
then
(((id Z) (#) (arccot * g)) + ((r / 2) (#) (ln * (f1 + f2)))) `| Z = f
by A10, PARTFUN1:5;
hence
integral (f,A) = ((((id Z) (#) (arccot * g)) + ((r / 2) (#) (ln * (f1 + f2)))) . (upper_bound A)) - ((((id Z) (#) (arccot * g)) + ((r / 2) (#) (ln * (f1 + f2)))) . (lower_bound A))
by A1, A5, A7, INTEGRA5:13; verum