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 = (exp_R * cosec) (#) (cos / (sin ^2)) & Z = dom f & f | A is continuous holds
integral (f,A) = ((- (exp_R * cosec)) . (upper_bound A)) - ((- (exp_R * cosec)) . (lower_bound A))
let f be PartFunc of REAL,REAL; for Z being open Subset of REAL st A c= Z & f = (exp_R * cosec) (#) (cos / (sin ^2)) & Z = dom f & f | A is continuous holds
integral (f,A) = ((- (exp_R * cosec)) . (upper_bound A)) - ((- (exp_R * cosec)) . (lower_bound A))
let Z be open Subset of REAL; ( A c= Z & f = (exp_R * cosec) (#) (cos / (sin ^2)) & Z = dom f & f | A is continuous implies integral (f,A) = ((- (exp_R * cosec)) . (upper_bound A)) - ((- (exp_R * cosec)) . (lower_bound A)) )
assume A1:
( A c= Z & f = (exp_R * cosec) (#) (cos / (sin ^2)) & Z = dom f & f | A is continuous )
; integral (f,A) = ((- (exp_R * cosec)) . (upper_bound A)) - ((- (exp_R * cosec)) . (lower_bound A))
then A2:
( f is_integrable_on A & f | A is bounded )
by INTEGRA5:10, INTEGRA5:11;
Z = (dom (exp_R * cosec)) /\ (dom (cos / (sin ^2)))
by A1, VALUED_1:def 4;
then A3:
( Z c= dom (exp_R * cosec) & Z c= dom (cos / (sin ^2)) )
by XBOOLE_1:18;
then A4:
- (exp_R * cosec) is_differentiable_on Z
by Th4;
A5:
for x being Real st x in Z holds
f . x = ((exp_R . (cosec . x)) * (cos . x)) / ((sin . x) ^2)
A7:
for x being Element of REAL st x in dom ((- (exp_R * cosec)) `| Z) holds
((- (exp_R * cosec)) `| Z) . x = f . x
dom ((- (exp_R * cosec)) `| Z) = dom f
by A1, A4, FDIFF_1:def 7;
then
(- (exp_R * cosec)) `| Z = f
by A7, PARTFUN1:5;
hence
integral (f,A) = ((- (exp_R * cosec)) . (upper_bound A)) - ((- (exp_R * cosec)) . (lower_bound A))
by A1, A2, A4, INTEGRA5:13; verum