Expectation of Integral of Brownian Motion

Question

I'm working through some stochastic analysis problems at the moment and I've come across a problem that is a bit tricky (to me) - does anyone know how to calculate this expecation? I'm not sure what to do with the exponential of the BM and the integral.

$\mathbb{E}_{\mathbb{P}} \bigg( \int_{0}^{t} exp(W_u) du \bigg) $

Thanks!

Answer 1

Hint: Apply Tonelli's (or Fubini's) theorem and use that

$$\mathbb{E}\exp(W_u) = \exp \left( \frac{1}{2} u \right)$$

since $W_u \sim N(0,u)$.