theorem :: INTEGR12:11

for A being non empty closed_interval Subset of REAL
for f 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
x > 0 ) & Z = dom f & f = (cos * ln) (#) ((id Z) ^) holds
integral (f,A) = ((sin * ln) . (upper_bound A)) - ((sin * ln) . (lower_bound A))