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 = (arccot / (id Z)) - (ln / (f1 + (#Z 2))) & Z c= ].(- 1),1.[ & Z = dom f & f | A is continuous holds
integral (f,A) = ((ln (#) arccot) . (upper_bound A)) - ((ln (#) arccot) . (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 = (arccot / (id Z)) - (ln / (f1 + (#Z 2))) & Z c= ].(- 1),1.[ & Z = dom f & f | A is continuous holds
integral (f,A) = ((ln (#) arccot) . (upper_bound A)) - ((ln (#) arccot) . (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 = (arccot / (id Z)) - (ln / (f1 + (#Z 2))) & Z c= ].(- 1),1.[ & Z = dom f & f | A is continuous implies integral (f,A) = ((ln (#) arccot) . (upper_bound A)) - ((ln (#) arccot) . (lower_bound A)) )
assume A1:
( A c= Z & ( for x being Real st x in Z holds
f1 . x = 1 ) & f = (arccot / (id Z)) - (ln / (f1 + (#Z 2))) & Z c= ].(- 1),1.[ & Z = dom f & f | A is continuous )
; integral (f,A) = ((ln (#) arccot) . (upper_bound A)) - ((ln (#) arccot) . (lower_bound A))
then A2:
( f is_integrable_on A & f | A is bounded )
by INTEGRA5:10, INTEGRA5:11;
Z = (dom (arccot / (id Z))) /\ (dom (ln / (f1 + (#Z 2))))
by A1, VALUED_1:12;
then A3:
( Z c= dom (arccot / (id Z)) & Z c= dom (ln / (f1 + (#Z 2))) )
by XBOOLE_1:18;
then
Z c= (dom arccot) /\ ((dom (id Z)) \ ((id Z) " {0}))
by RFUNCT_1:def 1;
then A4:
Z c= dom arccot
by XBOOLE_1:18;
Z c= (dom ln) /\ ((dom (f1 + (#Z 2))) \ ((f1 + (#Z 2)) " {0}))
by A3, RFUNCT_1:def 1;
then A5:
( Z c= dom ln & Z c= (dom (f1 + (#Z 2))) \ ((f1 + (#Z 2)) " {0}) )
by XBOOLE_1:18;
then
Z c= (dom arccot) /\ (dom ln)
by A4, XBOOLE_1:19;
then A6:
Z c= dom (ln (#) arccot)
by VALUED_1:def 4;
then A7:
ln (#) arccot is_differentiable_on Z
by A1, SIN_COS9:128;
A8:
Z c= dom ((f1 + (#Z 2)) ^)
by A5, RFUNCT_1:def 2;
dom ((f1 + (#Z 2)) ^) c= dom (f1 + (#Z 2))
by RFUNCT_1:1;
then A9:
Z c= dom (f1 + (#Z 2))
by A8;
A10:
for x being Real st x in Z holds
f . x = ((arccot . x) / x) - ((ln . x) / (1 + (x ^2)))
A12:
for x being Element of REAL st x in dom ((ln (#) arccot) `| Z) holds
((ln (#) arccot) `| Z) . x = f . x
dom ((ln (#) arccot) `| Z) = dom f
by A1, A7, FDIFF_1:def 7;
then
(ln (#) arccot) `| Z = f
by A12, PARTFUN1:5;
hence
integral (f,A) = ((ln (#) arccot) . (upper_bound A)) - ((ln (#) arccot) . (lower_bound A))
by A1, A2, A6, INTEGRA5:13, SIN_COS9:128; verum