theorem :: INTEGR14:53

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) ^2)) - (1 / ((sin . x) ^2)) ) & Z c= dom (tan + cot) & Z = dom f & f | A is continuous holds

integral (f,A) = ((tan + cot) . (upper_bound A)) - ((tan + cot) . (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) ^2)) - (1 / ((sin . x) ^2)) ) & Z c= dom (tan + cot) & Z = dom f & f | A is continuous holds

integral (f,A) = ((tan + cot) . (upper_bound A)) - ((tan + cot) . (lower_bound A))