Question Number 102296 by bemath last updated on 08/Jul/20

lim_(t→∞)  (1/t) ∫_0 ^t  sin (αx) cos (βx) dx

Answered by john santu last updated on 08/Jul/20

assume α,β > 0  lim_(t→∞) (1/t)∫_0 ^t sin (αx)cos (βx)dx  lim_(t→∞) (1/(2t))∫_0 ^t  sin (α+β)x+sin (α−β)x dx  lim_(t→∞)  −(1/(2t)) [((cos (α+β)x)/(α+β)) + ((cos (α−β)x)/(α−β)) ]_0 ^t   lim_(t→∞) ((cos (α+β)t)/(−2(α+β)t)) + lim_(t→∞) ((cos (α−β)t)/(−2(α−β)t)) = 0  (JS ⊛)