theorem :: INTEGR12:18

for r being Real
for A being non empty closed_interval Subset of REAL
for f, g, f1 being PartFunc of REAL,REAL
for Z being open Subset of REAL st A c= Z & f = (arccot * f1) - ((id Z) / (r (#) (g + (f1 ^2)))) & ( for x being Real st x in Z holds
( g . x = 1 & f1 . x = x / r & f1 . x > - 1 & f1 . x < 1 ) ) & Z = dom f & f | A is continuous holds
integral (f,A) = (((id Z) (#) (arccot * f1)) . (upper_bound A)) - (((id Z) (#) (arccot * f1)) . (lower_bound A))