theorem :: INTEGR11:46

for r being Real
for A being non empty closed_interval Subset of REAL
for Z being open Subset of REAL
for f being PartFunc of REAL,REAL st A c= Z & Z c= ].(- 1),1.[ & ( for x being Real st x in Z holds
f . x = r / (1 + (x ^2)) ) & Z = dom f & f | A is continuous holds
integral (f,A) = ((r (#) arctan) . (upper_bound A)) - ((r (#) arctan) . (lower_bound A))