with respect to the first equation i'm thinking the following:
focusing on X int Sin (kX) .dk
as the integral is in respect to k, simple integration gives
X *[1/X* -cos(kX)] = -cos(kX) as over 0_1
-cos(1X) + cos(0X) = 1 - cos(X)
Plugging in to formula
Y(t) = [1 - 1 - cos(X) ] * e^(t/2)
Y(t) = e^(t/2)cos(X)
as there is no drift term (.... .dt) this must mean its a martingale.
but i don't think i can perform simple integration on a function containing Brownian motion.