Let $f$ be a continuous function on $[0,\infty)$ and $B_t$ be a standard Brownian motion. Define
$X_t=\int_0^t f(s) dB(s).$
a) Show that $X_t$ is Gaussian and computer its covariance $C(X_s, X_t)$ b) prove that $e^{(X_t - 1/2 C(X_t, X_t))}$ is a continuous time martingale.
I think I know how to show $X_t$ is Gaussian (basically expressing it as a sum of Brownian motion increments each of which is gaussian) is that right? I don't know how to derive the covariance and show that $e^{(X_t - 1/2C(X_t, X_t))}$ is a martingale, any hint would be appreciated.