theorem :: INTEGR14:37

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

integral (f,A) = ((- (sec * cos)) . (upper_bound A)) - ((- (sec * cos)) . (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 = ((sin . x) * (sin . (cos . x))) / ((cos . (cos . x)) ^2) ) & Z c= dom (sec * cos) & Z = dom f & f | A is continuous holds

integral (f,A) = ((- (sec * cos)) . (upper_bound A)) - ((- (sec * cos)) . (lower_bound A))