let A be non empty 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 (cos,A) = - (2 * (sin 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 (cos,A) = - (2 * (sin x))
let n be Element of NAT ; :: thesis: ( A = [.(x + ((2 * n) * PI)),(x + (((2 * n) + 1) * PI)).] implies integral (cos,A) = - (2 * (sin x)) )
assume A = [.(x + ((2 * n) * PI)),(x + (((2 * n) + 1) * PI)).] ; :: thesis: integral (cos,A) = - (2 * (sin x))
then ( upper_bound A = x + (((2 * n) + 1) * PI) & lower_bound A = x + ((2 * n) * PI) ) by Th37;
then integral (cos,A) = (sin (x + (((2 * n) + 1) * PI))) - (sin (x + ((2 * n) * PI))) by Th39
.= (- (sin x)) - (sin (x + ((2 * n) * PI))) by Th2
.= (- (sin x)) - (sin x) by Th1
.= - (2 * (sin x)) ;
hence integral (cos,A) = - (2 * (sin x)) ; :: thesis: verum