I'm new to stochastic calculus, and I'm trying to evaluate: $$ \int_{0}^{1} W(r) f(r) dr, $$ and $$ \int_{0}^{t} W(r) f(r) dr, $$ where $f(t)$ is a deterministic square-integrable function and $W(t)$ is the standard brownian motion, with respect to time. For instance, we can consider a case where:
$$ \int_{0}^{1} W(r) \cos (2\pi r) dr $$ or for instance:
$$ \int_{0}^{1} W(r) e^{-r} dr $$
How could I use properties of the standard brownian motion to evaluate it?
If $f$ has an anti-derivative, say $F$, you can perform integration by parts: $$Z_t = \int_0^t W(r) f(r) dr = W(t) F(t) - \int_0^t F(s) dW_s = \int_0^t (F(t) - F(s)) dW_s$$
Since the Itô integral of a deterministic function is a Gaussian process, you can now fully identify $Z_t$ by computing its mean and covariance.