theorem :: INTEGR14:17

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

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

integral (f,A) = ((sec * exp_R) . (upper_bound A)) - ((sec * exp_R) . (lower_bound A))