theorem :: INTEGR13:8

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 & f = sin / exp_R & Z c= dom ((- (1 / 2)) (#) ((sin + cos) / exp_R)) & Z = dom f & f | A is continuous holds
integral (f,A) = (((- (1 / 2)) (#) ((sin + cos) / exp_R)) . (upper_bound A)) - (((- (1 / 2)) (#) ((sin + cos) / exp_R)) . (lower_bound A))