Expectation of an integral depending on Brownian motion

Question

I have a question which even if it involves a Brownian motion it shouldn't require too much effort.

Suppose I have a brownian Motion $W_t$ which has density $$p(t,x) := \frac{1}{\sqrt{2\pi t}} \exp \left(- \frac{x^2}{2t} \right), \qquad x \in \mathbb{R}$$

and I need to compute

$$ E[\int_0^t f'(W_r) dr]$$

where $f,f',f''$ are bounded and $f \in C^2$.

How can I move? I need to use the definition of expectation, and using the fact that I know the density of $W_r$, but the derivative confuses me a lot. I can't use Ito calculus

Answer 1

By Fubini's theorem, your integral equals $$\int_0^t E[f'(W_r)]\,dr$$ and since you know the density, you can compute $E[f'(W_r)]$ with an appropriate integral.