theorem :: INTEGR12:30

for a being Real
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 & Z c= dom (cot * f1) & f = ((cos * f1) ^2) / ((sin * f1) ^2) & ( for x being Real st x in Z holds
( f1 . x = a * x & a <> 0 ) ) & Z = dom f holds
integral (f,A) = ((((- (1 / a)) (#) (cot * f1)) - (id Z)) . (upper_bound A)) - ((((- (1 / a)) (#) (cot * f1)) - (id Z)) . (lower_bound A))