let r be Real; for A being non empty closed_interval Subset of REAL
for Z being open Subset of REAL
for f being PartFunc of REAL,REAL st A c= Z & Z c= ].(- 1),1.[ & ( for x being Real st x in Z holds
f . x = - (r / (1 + (x ^2))) ) & Z = dom f & f | A is continuous holds
integral (f,A) = ((r (#) arccot) . (upper_bound A)) - ((r (#) arccot) . (lower_bound A))
let A be non empty closed_interval Subset of REAL; for Z being open Subset of REAL
for f being PartFunc of REAL,REAL st A c= Z & Z c= ].(- 1),1.[ & ( for x being Real st x in Z holds
f . x = - (r / (1 + (x ^2))) ) & Z = dom f & f | A is continuous holds
integral (f,A) = ((r (#) arccot) . (upper_bound A)) - ((r (#) arccot) . (lower_bound A))
let Z be open Subset of REAL; for f being PartFunc of REAL,REAL st A c= Z & Z c= ].(- 1),1.[ & ( for x being Real st x in Z holds
f . x = - (r / (1 + (x ^2))) ) & Z = dom f & f | A is continuous holds
integral (f,A) = ((r (#) arccot) . (upper_bound A)) - ((r (#) arccot) . (lower_bound A))
let f be PartFunc of REAL,REAL; ( A c= Z & Z c= ].(- 1),1.[ & ( for x being Real st x in Z holds
f . x = - (r / (1 + (x ^2))) ) & Z = dom f & f | A is continuous implies integral (f,A) = ((r (#) arccot) . (upper_bound A)) - ((r (#) arccot) . (lower_bound A)) )
assume that
A1:
A c= Z
and
A2:
Z c= ].(- 1),1.[
and
A3:
for x being Real st x in Z holds
f . x = - (r / (1 + (x ^2)))
and
A4:
Z = dom f
and
A5:
f | A is continuous
; integral (f,A) = ((r (#) arccot) . (upper_bound A)) - ((r (#) arccot) . (lower_bound A))
A6:
r (#) arccot is_differentiable_on Z
by A2, SIN_COS9:84;
A7:
for x being Element of REAL st x in dom ((r (#) arccot) `| Z) holds
((r (#) arccot) `| Z) . x = f . x
dom ((r (#) arccot) `| Z) = dom f
by A4, A6, FDIFF_1:def 7;
then A9:
(r (#) arccot) `| Z = f
by A7, PARTFUN1:5;
( f is_integrable_on A & f | A is bounded )
by A1, A4, A5, INTEGRA5:10, INTEGRA5:11;
hence
integral (f,A) = ((r (#) arccot) . (upper_bound A)) - ((r (#) arccot) . (lower_bound A))
by A1, A2, A9, INTEGRA5:13, SIN_COS9:84; verum