Evaluating the integral of brownian motion with respect to time with defined limits of integration

Question

I am trying to evaluate the integral $$\int_0^1 W_t dt.$$

Using Ito's Lemma, I have got to the point where I can say

$$W_1 = \int_0^1 t dW_t + \int_0^1 W_t dt,$$ but this has not helped me evaluate the integral. Are there any methods for evaluating this limit or am I missing something obvious?