I'm studying from the book "Mathematical Analysis of Machine Learning Algorithms" by Tong Zhang. Theorem 2.9 states
Let $\{X_n\}_{n=1}^N$ be independent zero-mean sub-Gaussian random variables that satisfy $$ \ln\mathbb{E}[\exp(\lambda X_n)] \leq \frac{\lambda^2 b_n}{2}, $$ then for $\lambda < \frac{1}{2}b_n$ we have $$ \ln\mathbb{E}[\exp(\lambda X_n^2)] \leq -\frac{1}{2}\ln(1 - 2\lambda b_n) \tag{1} $$ [+ some other terms]
To prove (1) above, the author writes
proof Let $\xi\sim\mathcal{N}(0,1)$ which is independent of $X_n$. Then, for all $\lambda b_n <0.5$, we have $$ \begin{aligned} \Lambda_{X_n^2} &= \ln\mathbb{E}_{X_n}[\exp(\lambda X_n^2)]\\ &= \ln\mathbb{E}_{X_n}\mathbb{E}_\xi[\exp(\sqrt{2\lambda}\xi X_n)]\\ &= ... \end{aligned}\tag{2} $$
The second line in (2) it's what I'm interested in. I can't seem to understand how to transform a sub-Gaussian random variable to a Gaussian random variable. At the bottom of the proof the author writes: "The second [...] equality can be obtained using Gaussian integration", but the most I've been able to write is
$$ \ln\mathbb{E}_{X_n^{(1)}}\left[\int \exp\left( [\lambda^{1/2}X_n^{(1)}][\lambda^{1/2}X_n^{(2)}] \mathbb{1}(X_n^{(1)} - X_n^{(2)} =0) p(X_n^{(2)}) dX_n^{(2)}, \right)\right]. $$
I would appreciate any pointers.