theorem :: INTEGR14:20

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

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

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