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 = ((cos / sin) / (id Z)) - (ln / (sin ^2)) & Z c= dom (ln (#) cot) & Z = dom f & f | A is continuous holds
integral (f,A) = ((ln (#) cot) . (upper_bound A)) - ((ln (#) cot) . (lower_bound A))
let f be PartFunc of REAL,REAL; for Z being open Subset of REAL st A c= Z & f = ((cos / sin) / (id Z)) - (ln / (sin ^2)) & Z c= dom (ln (#) cot) & Z = dom f & f | A is continuous holds
integral (f,A) = ((ln (#) cot) . (upper_bound A)) - ((ln (#) cot) . (lower_bound A))
let Z be open Subset of REAL; ( A c= Z & f = ((cos / sin) / (id Z)) - (ln / (sin ^2)) & Z c= dom (ln (#) cot) & Z = dom f & f | A is continuous implies integral (f,A) = ((ln (#) cot) . (upper_bound A)) - ((ln (#) cot) . (lower_bound A)) )
assume A1:
( A c= Z & f = ((cos / sin) / (id Z)) - (ln / (sin ^2)) & Z c= dom (ln (#) cot) & Z = dom f & f | A is continuous )
; integral (f,A) = ((ln (#) cot) . (upper_bound A)) - ((ln (#) cot) . (lower_bound A))
then A2:
( f is_integrable_on A & f | A is bounded )
by INTEGRA5:10, INTEGRA5:11;
A3:
ln (#) cot is_differentiable_on Z
by A1, FDIFF_8:33;
Z = (dom ((cos / sin) / (id Z))) /\ (dom (ln / (sin ^2)))
by A1, VALUED_1:12;
then A4:
( Z c= dom ((cos / sin) / (id Z)) & Z c= dom (ln / (sin ^2)) )
by XBOOLE_1:18;
dom ((cos / sin) / (id Z)) c= (dom (cos / sin)) /\ ((dom (id Z)) \ ((id Z) " {0}))
by RFUNCT_1:def 1;
then
dom ((cos / sin) / (id Z)) c= dom (cos / sin)
by XBOOLE_1:18;
then A5:
Z c= dom (cos / sin)
by A4;
A6:
for x being Real st x in Z holds
f . x = (((cos . x) / (sin . x)) / x) - ((ln . x) / ((sin . x) ^2))
A8:
for x being Element of REAL st x in dom ((ln (#) cot) `| Z) holds
((ln (#) cot) `| Z) . x = f . x
dom ((ln (#) cot) `| Z) = dom f
by A1, A3, FDIFF_1:def 7;
then
(ln (#) cot) `| Z = f
by A8, PARTFUN1:5;
hence
integral (f,A) = ((ln (#) cot) . (upper_bound A)) - ((ln (#) cot) . (lower_bound A))
by A1, A2, FDIFF_8:33, INTEGRA5:13; verum