theorem :: INTEGR14:15

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 . (1 / x)) / ((x ^2) * ((cos . (1 / x)) ^2)) ) & Z c= dom (sec * ((id Z) ^)) & Z = dom f & f | A is continuous holds

integral (f,A) = ((- (sec * ((id Z) ^))) . (upper_bound A)) - ((- (sec * ((id Z) ^))) . (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 . (1 / x)) / ((x ^2) * ((cos . (1 / x)) ^2)) ) & Z c= dom (sec * ((id Z) ^)) & Z = dom f & f | A is continuous holds

integral (f,A) = ((- (sec * ((id Z) ^))) . (upper_bound A)) - ((- (sec * ((id Z) ^))) . (lower_bound A))