theorem :: INTEGR12:41

for n being Element of 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 & n > 0 & ( for x being Real st x in Z holds
( f . x = (cos . x) / ((sin . x) #Z (n + 1)) & sin . x <> 0 ) ) & Z c= dom ((#Z n) * (sin ^)) & Z = dom f & f | A is continuous holds
integral (f,A) = (((- (1 / n)) (#) ((#Z n) * (sin ^))) . (upper_bound A)) - (((- (1 / n)) (#) ((#Z n) * (sin ^))) . (lower_bound A))