theorem :: INTEGR14:27

for n being Nat

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 = (n * (cos . x)) / ((sin . x) #Z (n + 1)) ) & Z c= dom ((#Z n) * cosec) & 1 <= n & Z = dom f & f | A is continuous holds

integral (f,A) = ((- ((#Z n) * cosec)) . (upper_bound A)) - ((- ((#Z n) * cosec)) . (lower_bound A))

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 = (n * (cos . x)) / ((sin . x) #Z (n + 1)) ) & Z c= dom ((#Z n) * cosec) & 1 <= n & Z = dom f & f | A is continuous holds

integral (f,A) = ((- ((#Z n) * cosec)) . (upper_bound A)) - ((- ((#Z n) * cosec)) . (lower_bound A))