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 ((sin + cos),A) = (2 * (cos x)) - (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 ((sin + cos),A) = (2 * (cos x)) - (2 * (sin x))
let n be Element of NAT ; :: thesis: ( A = [.(x + ((2 * n) * PI)),(x + (((2 * n) + 1) * PI)).] implies integral ((sin + cos),A) = (2 * (cos x)) - (2 * (sin x)) )
assume A = [.(x + ((2 * n) * PI)),(x + (((2 * n) + 1) * PI)).] ; :: thesis: integral ((sin + cos),A) = (2 * (cos x)) - (2 * (sin x))
then ( upper_bound A = x + (((2 * n) + 1) * PI) & lower_bound A = x + ((2 * n) * PI) ) by Th37;
then integral ((sin + cos),A) = ((((- cos) . (x + (((2 * n) + 1) * PI))) - ((- cos) . (x + ((2 * n) * PI)))) + (sin . (x + (((2 * n) + 1) * PI)))) - (sin . (x + ((2 * n) * PI))) by Th69
.= (((- (cos . (x + (((2 * n) + 1) * PI)))) - ((- cos) . (x + ((2 * n) * PI)))) + (sin . (x + (((2 * n) + 1) * PI)))) - (sin . (x + ((2 * n) * PI))) by VALUED_1:8
.= (((- (cos (x + (((2 * n) + 1) * PI)))) - (- (cos (x + ((2 * n) * PI))))) + (sin (x + (((2 * n) + 1) * PI)))) - (sin (x + ((2 * n) * PI))) by VALUED_1:8
.= (((- (- (cos x))) - (- (cos (x + ((2 * n) * PI))))) + (sin (x + (((2 * n) + 1) * PI)))) - (sin (x + ((2 * n) * PI))) by Th4
.= (((- (- (cos x))) - (- (cos x))) + (sin (x + (((2 * n) + 1) * PI)))) - (sin (x + ((2 * n) * PI))) by Th3
.= (((- (- (cos x))) - (- (cos x))) + (- (sin x))) - (sin (x + ((2 * n) * PI))) by Th2
.= (((cos x) + (cos x)) - (sin x)) - (sin x) by Th1
.= (2 * (cos x)) - (2 * (sin x)) ;
hence integral ((sin + cos),A) = (2 * (cos x)) - (2 * (sin x)) ; :: thesis: verum