theorem :: INTEGR14:43

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 . (cot . x)) / ((sin . x) ^2)) / ((sin . (cot . x)) ^2) ) & Z c= dom (cosec * cot) & Z = dom f & f | A is continuous holds

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

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