theorem :: INTEGR12:43

for A being non empty closed_interval Subset of REAL
for f, f2, g1, g2 being PartFunc of REAL,REAL
for Z being open Subset of REAL st A c= Z & f = ((g1 + g2) ^) / f2 & f2 = arccot & Z c= ].(- 1),1.[ & g2 = #Z 2 & ( for x being Real st x in Z holds
( f . x = 1 / ((1 + (x ^2)) * (arccot . x)) & g1 . x = 1 & f2 . x > 0 ) ) & Z = dom f holds
integral (f,A) = ((- (ln * arccot)) . (upper_bound A)) - ((- (ln * arccot)) . (lower_bound A))