theorem :: INTEGR14:54

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 . (sin . x)) * (cos . x) ) & Z = dom f & f | A is continuous holds

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

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