:: Integrability Formulas -- Part {I}
:: by Bo Li and Na Ma
::
:: Copyright (c) 2009-2021 Association of Mizar Users

theorem Th1: :: INTEGR12:1
for f1, f2 being PartFunc of REAL,REAL
for Z being open Subset of REAL st Z c= dom ((f1 + f2) ^) & ( for x being Real st x in Z holds
f1 . x = 1 ) & f2 = #Z 2 holds
( (f1 + f2) ^ is_differentiable_on Z & ( for x being Real st x in Z holds
(((f1 + f2) ^) `| Z) . x = - ((2 * x) / ((1 + (x |^ 2)) ^2)) ) )
proof end;

::f.x=1/((1+x^2)*arccot.x)
theorem :: INTEGR12:2
for A being non empty closed_interval Subset of REAL
for f, f2, g1, g2 being PartFunc of REAL,REAL
for Z being open Subset of REAL st A c= Z & f = ((g1 + g2) ^) / f2 & f2 = arccot & Z c= ].(- 1),1.[ & g2 = #Z 2 & ( for x being Real st x in Z holds
( g1 . x = 1 & f2 . x > 0 ) ) & Z = dom f holds
integral (f,A) = ((- ()) . ()) - ((- ()) . ())
proof end;

::f.x=exp_R.x/(1+(exp_R.x)^2)
theorem :: INTEGR12:3
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 & ( for x being Real st x in Z holds
( exp_R . x < 1 & f1 . x = 1 ) ) & Z c= dom () & Z = dom f & f = exp_R / (f1 + ()) holds
integral (f,A) = (() . ()) - (() . ())
proof end;

::f.x=-exp_R.x/(1+(exp_R.x)^2)
theorem :: INTEGR12:4
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 & ( for x being Real st x in Z holds
( exp_R . x < 1 & f1 . x = 1 ) ) & Z c= dom () & Z = dom f & f = () / (f1 + ()) holds
integral (f,A) = (() . ()) - (() . ())
proof end;

::f.x=exp_R.x*sin.x/cos.x+exp_R.x/(cos.x)^2
theorem :: INTEGR12:5
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 & Z = dom f & f = (exp_R (#) ()) + (exp_R / ()) holds
integral (f,A) = (() . ()) - (() . ())
proof end;

::f.x=exp_R.x*cos.x/sin.x-exp_R.x/(sin.x)^2
theorem :: INTEGR12:6
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 & Z = dom f & f = (exp_R (#) ()) - (exp_R / ()) holds
integral (f,A) = (() . ()) - (() . ())
proof end;

::f.x=exp_R.x*arctan.x+exp_R.x/(1+x^2)
theorem :: INTEGR12:7
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 & ( for x being Real st x in Z holds
f1 . x = 1 ) & Z c= ].(- 1),1.[ & Z = dom f & f = () + (exp_R / (f1 + (#Z 2))) holds
integral (f,A) = (() . ()) - (() . ())
proof end;

::f.x=exp_R.x*arccot.x-exp_R.x/(1+x^2)
theorem :: INTEGR12:8
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 & ( for x being Real st x in Z holds
f1 . x = 1 ) & Z c= ].(- 1),1.[ & Z = dom f & f = () - (exp_R / (f1 + (#Z 2))) holds
integral (f,A) = (() . ()) - (() . ())
proof end;

::f.x=exp_R.(sin.x) * cos.x
theorem :: INTEGR12:9
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 & Z = dom f & f = () (#) cos holds
integral (f,A) = (() . ()) - (() . ())
proof end;

::f.x=exp_R.(cos.x) * sin.x
theorem :: INTEGR12:10
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 & Z = dom f & f = () (#) sin holds
integral (f,A) = ((- ()) . ()) - ((- ()) . ())
proof end;

::f.x=cos.(ln.x)/x
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 = () (#) ((id Z) ^) holds
integral (f,A) = (() . ()) - (() . ())
proof end;

Lm1: right_open_halfline 0 = { g where g is Real : 0 < g }
by XXREAL_1:230;

::f.x=sin.(ln.x)/x
theorem :: INTEGR12:12
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 = () (#) ((id Z) ^) holds
integral (f,A) = ((- ()) . ()) - ((- ()) . ())
proof end;

::f.x=exp_R.x * cos.(exp_R.x)
theorem :: INTEGR12:13
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 & Z = dom f & f = exp_R (#) () holds
integral (f,A) = (() . ()) - (() . ())
proof end;

::f.x = exp_R.x * sin.(exp_R.x)
theorem :: INTEGR12:14
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 & Z = dom f & f = exp_R (#) () holds
integral (f,A) = ((- ()) . ()) - ((- ()) . ())
proof end;

::f.x=arctan.(x/r)
theorem :: INTEGR12:15
for r being Real
for A being non empty closed_interval Subset of REAL
for f, g, f1, f2 being PartFunc of REAL,REAL
for Z being open Subset of REAL st A c= Z & Z c= dom (ln * (f1 + f2)) & r <> 0 & ( for x being Real st x in Z holds
( g . x = x / r & g . x > - 1 & g . x < 1 & f1 . x = 1 ) ) & f2 = (#Z 2) * g & Z = dom f & f = arctan * g holds
integral (f,A) = ((((id Z) (#) ()) - ((r / 2) (#) (ln * (f1 + f2)))) . ()) - ((((id Z) (#) ()) - ((r / 2) (#) (ln * (f1 + f2)))) . ())
proof end;

::f.x=arccot.(x/r)
theorem :: INTEGR12:16
for r being Real
for A being non empty closed_interval Subset of REAL
for f, g, f1, f2 being PartFunc of REAL,REAL
for Z being open Subset of REAL st A c= Z & Z c= dom (ln * (f1 + f2)) & r <> 0 & ( for x being Real st x in Z holds
( g . x = x / r & g . x > - 1 & g . x < 1 & f1 . x = 1 ) ) & f2 = (#Z 2) * g & Z = dom f & f = arccot * g holds
integral (f,A) = ((((id Z) (#) ()) + ((r / 2) (#) (ln * (f1 + f2)))) . ()) - ((((id Z) (#) ()) + ((r / 2) (#) (ln * (f1 + f2)))) . ())
proof end;

::f.x=arctan.(x/r)+x/(r*(1+(x/r)^2))
theorem :: INTEGR12:17
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 = (arctan * 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) (#) (arctan * f1)) . ()) - (((id Z) (#) (arctan * f1)) . ())
proof end;

::f.x=arccot.(x/r)-x/(r*(1+(x/r)^2))
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)) . ()) - (((id Z) (#) (arccot * f1)) . ())
proof end;

::f.x=n*(arcsin.x) #Z (n-1) / sqrt(1-x^2)
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) . ()) - (((#Z n) * arcsin) . ())
proof end;

::f.x=n*(arccos.x) #Z (n-1) / sqrt(1-x^2)
theorem :: INTEGR12:20
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 c= dom ((#Z n) * arccos) & Z = dom f & 1 < n & f = (n (#) ((#Z (n - 1)) * arccos)) / ((#R (1 / 2)) * (f1 - (#Z 2))) holds
integral (f,A) = ((- ((#Z n) * arccos)) . ()) - ((- ((#Z n) * arccos)) . ())
proof end;

::f.x=arcsin.x+x/sqrt(1-x^2)
theorem :: INTEGR12:21
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 & ( for x being Real st x in Z holds
f1 . x = 1 ) & Z c= ].(- 1),1.[ & Z = dom f & f = arcsin + ((id Z) / ((#R (1 / 2)) * (f1 - (#Z 2)))) holds
integral (f,A) = (((id Z) (#) arcsin) . ()) - (((id Z) (#) arcsin) . ())
proof end;

::f.x=arccos.x-x/sqrt(1-x^2)
theorem :: INTEGR12:22
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 & ( for x being Real st x in Z holds
f1 . x = 1 ) & Z c= ].(- 1),1.[ & Z = dom f & f = arccos - ((id Z) / ((#R (1 / 2)) * (f1 - (#Z 2)))) holds
integral (f,A) = (((id Z) (#) arccos) . ()) - (((id Z) (#) arccos) . ())
proof end;

::f.x=a*arcsin.x+(a*x+b)/sqrt(1-x^2)
theorem :: INTEGR12:23
for a, b being Real
for A being non empty closed_interval Subset of REAL
for f, f1, f2 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 = (a * x) + b & f2 . x = 1 ) ) & Z = dom f & f = () + (f1 / ((#R (1 / 2)) * (f2 - (#Z 2)))) holds
integral (f,A) = (() . ()) - (() . ())
proof end;

::f.x=a*arccos.x-(a*x+b)/sqrt(1-x^2)
theorem :: INTEGR12:24
for a, b being Real
for A being non empty closed_interval Subset of REAL
for f, f1, f2 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 = (a * x) + b & f2 . x = 1 ) ) & Z = dom f & f = () - (f1 / ((#R (1 / 2)) * (f2 - (#Z 2)))) holds
integral (f,A) = (() . ()) - (() . ())
proof end;

::f.x=arcsin.(x/a)+x/(a*sqrt(1-(x/a)^2))
theorem :: INTEGR12:25
for a 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 & ( for x being Real st x in Z holds
( g . x = 1 & f1 . x = x / a & f1 . x > - 1 & f1 . x < 1 ) ) & Z = dom f & f | A is continuous & f = (arcsin * f1) + ((id Z) / (a (#) ((#R (1 / 2)) * (g - (f1 ^2))))) holds
integral (f,A) = (((id Z) (#) (arcsin * f1)) . ()) - (((id Z) (#) (arcsin * f1)) . ())
proof end;

::f.x=arccos.(x/a)-x/(a*sqrt(1-(x/a)^2))
theorem :: INTEGR12:26
for a 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 & ( for x being Real st x in Z holds
( g . x = 1 & f1 . x = x / a & f1 . x > - 1 & f1 . x < 1 ) ) & Z = dom f & f | A is continuous & f = (arccos * f1) - ((id Z) / (a (#) ((#R (1 / 2)) * (g - (f1 ^2))))) holds
integral (f,A) = (((id Z) (#) (arccos * f1)) . ()) - (((id Z) (#) (arccos * f1)) . ())
proof end;

::f.x = n*(sin.x) #Z (n-1)/(cos.x) #Z (n+1)
theorem :: INTEGR12:27
for n being Element of NAT
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 & f = (n (#) ((#Z (n - 1)) * sin)) / ((#Z (n + 1)) * cos) & 1 <= n & Z c= dom ((#Z n) * tan) & Z = dom f holds
integral (f,A) = (((#Z n) * tan) . ()) - (((#Z n) * tan) . ())
proof end;

::f.x=n*(cos.x) #Z (n-1)/(sin.x) #Z (n+1)
theorem :: INTEGR12:28
for n being Element of NAT
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 & f = (n (#) ((#Z (n - 1)) * cos)) / ((#Z (n + 1)) * sin) & 1 <= n & Z c= dom ((#Z n) * cot) & Z = dom f holds
integral (f,A) = ((- ((#Z n) * cot)) . ()) - ((- ((#Z n) * cot)) . ())
proof end;

::f.x=(sin.(a*x))^2/(cos.(a*x))^2
theorem :: INTEGR12:29
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 (tan * f1) & f = ((sin * f1) ^2) / ((cos * 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) (#) (tan * f1)) - (id Z)) . ()) - ((((1 / a) (#) (tan * f1)) - (id Z)) . ())
proof end;

::f.x=(cos.(a*x))^2/(sin.(a*x))^2
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)) . ()) - ((((- (1 / a)) (#) (cot * f1)) - (id Z)) . ())
proof end;

::f.x=a*sin.x/cos.x+(a*x+b)/(cos.x)^2
theorem :: INTEGR12:31
for a, b 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 & ( for x being Real st x in Z holds
f1 . x = (a * x) + b ) & Z = dom f & f = (a (#) ()) + (f1 / ()) holds
integral (f,A) = ((f1 (#) tan) . ()) - ((f1 (#) tan) . ())
proof end;

::f.x=a*cos.x/sin.x-(a*x+b)/(sin.x)^2
theorem :: INTEGR12:32
for a, b 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 & ( for x being Real st x in Z holds
f1 . x = (a * x) + b ) & Z = dom f & f = (a (#) ()) - (f1 / ()) holds
integral (f,A) = ((f1 (#) cot) . ()) - ((f1 (#) cot) . ())
proof end;

::f.x=(sin.x)^2/(cos.x)^2
theorem :: INTEGR12:33
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
f . x = ((sin . x) ^2) / ((cos . x) ^2) ) & Z c= dom (tan - (id Z)) & Z = dom f & f | A is continuous holds
integral (f,A) = ((tan - (id Z)) . ()) - ((tan - (id Z)) . ())
proof end;

::f.x=(cos.x)^2/(sin.x)^2
theorem :: INTEGR12:34
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
f . x = ((cos . x) ^2) / ((sin . x) ^2) ) & Z c= dom (() - (id Z)) & Z = dom f & f | A is continuous holds
integral (f,A) = ((() - (id Z)) . ()) - ((() - (id Z)) . ())
proof end;

::f.x=1/(x*(1+(ln.x)^2))
theorem :: INTEGR12:35
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
( f . x = 1 / (x * (1 + ((ln . x) ^2))) & ln . x > - 1 & ln . x < 1 ) ) & Z c= dom () & Z = dom f & f | A is continuous holds
integral (f,A) = (() . ()) - (() . ())
proof end;

::f.x=-1/(x*(1+(ln.x)^2))
theorem :: INTEGR12:36
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
( f . x = - (1 / (x * (1 + ((ln . x) ^2)))) & ln . x > - 1 & ln . x < 1 ) ) & Z c= dom () & Z = dom f & f | A is continuous holds
integral (f,A) = (() . ()) - (() . ())
proof end;

::f.x=a / sqrt(1-(a*x+b)^2)
theorem :: INTEGR12:37
for a, b 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 & ( for x being Real st x in Z holds
( f . x = a / (sqrt (1 - (((a * x) + b) ^2))) & f1 . x = (a * x) + b & f1 . x > - 1 & f1 . x < 1 ) ) & Z c= dom (arcsin * f1) & Z = dom f & f | A is continuous holds
integral (f,A) = ((arcsin * f1) . ()) - ((arcsin * f1) . ())
proof end;

theorem :: INTEGR12:38
for a, b 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 & ( for x being Real st x in Z holds
( f . x = a / (sqrt (1 - (((a * x) + b) ^2))) & f1 . x = (a * x) + b & f1 . x > - 1 & f1 . x < 1 ) ) & Z c= dom (arccos * f1) & Z = dom f & f | A is continuous holds
integral (f,A) = ((- (arccos * f1)) . ()) - ((- (arccos * f1)) . ())
proof end;

::f.x=x*(1-x #Z 2) #R (-1/2)
theorem :: INTEGR12:39
for A being non empty closed_interval Subset of REAL
for f, g, f1, f2 being PartFunc of REAL,REAL
for Z being open Subset of REAL st A c= Z & f1 = g - f2 & f2 = #Z 2 & ( for x being Real st x in Z holds
( f . x = x * ((1 - (x #Z 2)) #R (- (1 / 2))) & g . x = 1 & f1 . x > 0 ) ) & Z c= dom ((#R (1 / 2)) * f1) & Z = dom f & f | A is continuous holds
integral (f,A) = ((- ((#R (1 / 2)) * f1)) . ()) - ((- ((#R (1 / 2)) * f1)) . ())
proof end;

::f.x=x*(a^2-x #Z 2) #R (-1/2)
theorem :: INTEGR12:40
for a being Real
for A being non empty closed_interval Subset of REAL
for f, g, f1, f2 being PartFunc of REAL,REAL
for Z being open Subset of REAL st A c= Z & g = f1 - f2 & f2 = #Z 2 & ( for x being Real st x in Z holds
( f . x = x * (((a ^2) - (x #Z 2)) #R (- (1 / 2))) & f1 . x = a ^2 & g . x > 0 ) ) & Z c= dom ((#R (1 / 2)) * g) & Z = dom f & f | A is continuous holds
integral (f,A) = ((- ((#R (1 / 2)) * g)) . ()) - ((- ((#R (1 / 2)) * g)) . ())
proof end;

::f.x=cos.x/((sin.x) #Z (n+1))
theorem :: INTEGR12:41
for n being Element of NAT
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 & n > 0 & ( for x being Real st x in Z holds
( f . x = (cos . x) / ((sin . x) #Z (n + 1)) & sin . x <> 0 ) ) & Z c= dom ((#Z n) * ()) & Z = dom f & f | A is continuous holds
integral (f,A) = (((- (1 / n)) (#) ((#Z n) * ())) . ()) - (((- (1 / n)) (#) ((#Z n) * ())) . ())
proof end;

::f.x=sin.x/((cos.x) #Z (n+1))
theorem :: INTEGR12:42
for n being Element of NAT
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 & n > 0 & ( for x being Real st x in Z holds
( f . x = (sin . x) / ((cos . x) #Z (n + 1)) & cos . x <> 0 ) ) & Z c= dom ((#Z n) * ()) & Z = dom f & f | A is continuous holds
integral (f,A) = (((1 / n) (#) ((#Z n) * ())) . ()) - (((1 / n) (#) ((#Z n) * ())) . ())
proof end;

::f.x=1/((1+x^2)*arccot.x)
theorem :: INTEGR12:43
for A being non empty closed_interval Subset of REAL
for f, f2, g1, g2 being PartFunc of REAL,REAL
for Z being open Subset of REAL st A c= Z & f = ((g1 + g2) ^) / f2 & f2 = arccot & Z c= ].(- 1),1.[ & g2 = #Z 2 & ( for x being Real st x in Z holds
( f . x = 1 / ((1 + (x ^2)) * ()) & g1 . x = 1 & f2 . x > 0 ) ) & Z = dom f holds
integral (f,A) = ((- ()) . ()) - ((- ()) . ())
proof end;

::f.x=1 / (sqrt(1-x^2)*arcsin.x)
theorem :: INTEGR12:44
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
( arcsin . x > 0 & f1 . x = 1 ) ) & Z c= dom () & Z = dom f & f = (((#R (1 / 2)) * (f1 - (#Z 2))) (#) arcsin) ^ holds
integral (f,A) = (() . ()) - (() . ())
proof end;

::f.x=1 / (sqrt(1-x^2)*arccos.x)
theorem :: INTEGR12:45
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 & arccos . x > 0 ) ) & Z c= dom () & Z = dom f & f = (((#R (1 / 2)) * (f1 - (#Z 2))) (#) arccos) ^ holds
integral (f,A) = ((- ()) . ()) - ((- ()) . ())
proof end;