theorem :: INTEGR14:32

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

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

integral (f,A) = ((ln (#) sec) . (upper_bound A)) - ((ln (#) sec) . (lower_bound A))