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I am trying to compute the following covariance

$Cov$ $(\int^t_0 X(u)du$,$\int^t_0 X(s)ds$)

with $X(u)$= $e^{-at}$$(X(0) + \int^t_0$$\sigma$$e^{au}dW_u$)

Can I use the covariance of Ito Gaussian integrals ?

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    note that $\int_0^t X(u)\;du = \int_0^t X(s)\;ds$. so basically you are looking for the variance of $\int_0^t X(u)\;du$ – 2017-01-12

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Note that \begin{align*} \int_0^tX_sds &= X_0\int_0^t e^{-as}ds + \int_0^t \int_0^s \sigma e^{-as}e^{au} dW_u ds\\ &=X_0\int_0^t e^{-as}ds + \int_0^t \sigma e^{au}\left(\int_u^t e^{-as} ds\right) dW_u\\ &=X_0\int_0^t e^{-as}ds + \int_0^t\frac{\sigma}{a} \left(1-e^{-a(t-u)}\right)dW_u. \end{align*} The remaining should now be straightforward.