I was studying on the Bergomi volatility model(using forward variance represented as $\xi_{t}^{T}$).However I don't understand how the author passes from the sde to the first step by only integrating respectively over $\xi_{t}^{T}$ and $W_{t}^{k}$.
$\begin{array}{l} \text {The dynamics are represented as : }\left\{\begin{array}{l} d S_{t}^{\omega}=(r-q) S_{t}^{\omega} d t+\sqrt{\xi_{t}^{t}} S_{t}^{\omega} d Z_{t} \\ d \xi_{t}^{T}=\omega \xi_{t}^{T} \sum_{k} \lambda_{k t}^{T}\left(\xi_{t}\right) d W_{t}^{k} \end{array}\right.\\ \text { At order } 1(\text { Using one factor}) \text { in } \omega: \quad \xi_{t}^{T}=\xi_{0}^{T}\left(1+\omega \int_{0}^{t} \sum_{k}\left(\lambda_{k \tau}^{T}\right)_{0} d W_{\tau}^{k}\right) \end{array}$
With The instantaneous variance of the spot process such $\xi_{t}^{t}$, $S_t$ the stock price, $w$ a scaling factor,$d W_{\tau}^{k}$ correlated standard brownian motions.
Could someone help me to understand this step thank you.