theorem :: INTEGR14:59

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