theorem :: INTEGR11:52

for A being non empty closed_interval Subset of REAL
for Z being open Subset of REAL
for f being PartFunc of REAL,REAL st A c= Z & ( for x being Real st x in Z holds
( 1 + (cos . x) <> 0 & 1 - (cos . x) <> 0 & f . x = (cos . x) / ((cos . x) - 1) ) ) & dom (((id Z) + cot) + cosec) = Z & Z = dom f & f | A is continuous holds
integral (f,A) = ((((id Z) + cot) + cosec) . (upper_bound A)) - ((((id Z) + cot) + cosec) . (lower_bound A))