theorem :: INTEGR14:38

for A being 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 & ( for x being Real st x in Z holds

f . x = ((cos . x) * (cos . (sin . x))) / ((sin . (sin . x)) ^2) ) & Z c= dom (cosec * sin) & Z = dom f & f | A is continuous holds

integral (f,A) = ((- (cosec * sin)) . (upper_bound A)) - ((- (cosec * sin)) . (lower_bound A))

for f 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

f . x = ((cos . x) * (cos . (sin . x))) / ((sin . (sin . x)) ^2) ) & Z c= dom (cosec * sin) & Z = dom f & f | A is continuous holds

integral (f,A) = ((- (cosec * sin)) . (upper_bound A)) - ((- (cosec * sin)) . (lower_bound A))