theorem :: INTEGR12:19

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 & Z c= ].(- 1),1.[ & ( for x being Real st x in Z holds
f1 . x = 1 ) & Z = dom f & Z c= dom ((#Z n) * arcsin) & 1 < n & f = (n (#) ((#Z (n - 1)) * arcsin)) / ((#R (1 / 2)) * (f1 - (#Z 2))) holds
integral (f,A) = (((#Z n) * arcsin) . (upper_bound A)) - (((#Z n) * arcsin) . (lower_bound A))