let A be closed-interval Subset of REAL ; :: thesis: for x being Real
for n being Element of NAT st A = [.(x + ((2 * n) * PI )),(x + (((2 * n) + 1) * PI )).] holds
integral (- sin ),A = - (2 * (cos x))
let x be Real; :: thesis: for n being Element of NAT st A = [.(x + ((2 * n) * PI )),(x + (((2 * n) + 1) * PI )).] holds
integral (- sin ),A = - (2 * (cos x))
let n be Element of NAT ; :: thesis: ( A = [.(x + ((2 * n) * PI )),(x + (((2 * n) + 1) * PI )).] implies integral (- sin ),A = - (2 * (cos x)) )
assume A = [.(x + ((2 * n) * PI )),(x + (((2 * n) + 1) * PI )).] ; :: thesis: integral (- sin ),A = - (2 * (cos x))
then ( upper_bound A = x + (((2 * n) + 1) * PI ) & lower_bound A = x + ((2 * n) * PI ) ) by Th37;
then integral (- sin ),A = (cos (x + (((2 * n) + 1) * PI ))) - (cos (x + ((2 * n) * PI ))) by Th46
.= (- (cos x)) - (cos (x + ((2 * n) * PI ))) by Th4
.= (- (cos x)) - (cos x) by Th3
.= - (2 * (cos x)) ;
hence integral (- sin ),A = - (2 * (cos x)) ; :: thesis: verum