theorem :: INTEGR11:72

for A being non empty closed_interval Subset of REAL
for Z being open Subset of REAL
for f1, f2 being PartFunc of REAL,REAL st A c= Z & Z c= ].(- 1),1.[ & f2 = #Z 2 & ( for x being Real st x in Z holds
f1 . x = 1 ) & Z = dom arctan & Z = dom (((id Z) (#) arctan) - ((1 / 2) (#) (ln * (f1 + f2)))) holds
integral (arctan,A) = ((((id Z) (#) arctan) - ((1 / 2) (#) (ln * (f1 + f2)))) . (upper_bound A)) - ((((id Z) (#) arctan) - ((1 / 2) (#) (ln * (f1 + f2)))) . (lower_bound A))