let A be non empty closed_interval Subset of REAL; for f being PartFunc of REAL,REAL
for Z being open Subset of REAL st A c= Z & f = ((id Z) (#) ((cos * ln) ^2)) ^ & Z c= dom (tan * ln) & Z = dom f & f | A is continuous holds
integral (f,A) = ((tan * ln) . (upper_bound A)) - ((tan * ln) . (lower_bound A))
let f be PartFunc of REAL,REAL; for Z being open Subset of REAL st A c= Z & f = ((id Z) (#) ((cos * ln) ^2)) ^ & Z c= dom (tan * ln) & Z = dom f & f | A is continuous holds
integral (f,A) = ((tan * ln) . (upper_bound A)) - ((tan * ln) . (lower_bound A))
let Z be open Subset of REAL; ( A c= Z & f = ((id Z) (#) ((cos * ln) ^2)) ^ & Z c= dom (tan * ln) & Z = dom f & f | A is continuous implies integral (f,A) = ((tan * ln) . (upper_bound A)) - ((tan * ln) . (lower_bound A)) )
assume A1:
( A c= Z & f = ((id Z) (#) ((cos * ln) ^2)) ^ & Z c= dom (tan * ln) & Z = dom f & f | A is continuous )
; integral (f,A) = ((tan * ln) . (upper_bound A)) - ((tan * ln) . (lower_bound A))
then A2:
( f is_integrable_on A & f | A is bounded )
by INTEGRA5:10, INTEGRA5:11;
A3:
tan * ln is_differentiable_on Z
by A1, FDIFF_8:14;
Z c= dom ((id Z) (#) ((cos * ln) ^2))
by A1, RFUNCT_1:1;
then
Z c= (dom (id Z)) /\ (dom ((cos * ln) ^2))
by VALUED_1:def 4;
then
Z c= dom ((cos * ln) ^2)
by XBOOLE_1:18;
then A4:
Z c= dom (cos * ln)
by VALUED_1:11;
A5:
for x being Real st x in Z holds
f . x = 1 / (x * ((cos . (ln . x)) ^2))
A7:
for x being Element of REAL st x in dom ((tan * ln) `| Z) holds
((tan * ln) `| Z) . x = f . x
dom ((tan * ln) `| Z) = dom f
by A1, A3, FDIFF_1:def 7;
then
(tan * ln) `| Z = f
by A7, PARTFUN1:5;
hence
integral (f,A) = ((tan * ln) . (upper_bound A)) - ((tan * ln) . (lower_bound A))
by A1, A2, FDIFF_8:14, INTEGRA5:13; verum