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 - 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 Th78
.= ((- (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