theorem :: INTEGR13:39

for n being Element of NAT
for A being non empty closed_interval Subset of REAL
for f, f1 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
f1 . x = 1 ) & f = - (n (#) (((#Z (n - 1)) * arccot) / (f1 + (#Z 2)))) & Z c= ].(- 1),1.[ & Z c= dom ((#Z n) * arccot) & Z = dom f & f | A is continuous holds
integral (f,A) = (((#Z n) * arccot) . (upper_bound A)) - (((#Z n) * arccot) . (lower_bound A))