theorem :: INTEGR14:47

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

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

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