Proof of $\mathbb E[\exp( \lambda XY)]=\mathbb E[\exp( \lambda^2 X^2)/2]$, where $X,Y$ are independent standard normal random variables
Write $$\begin{align} E(\exp(\lambda XY)) &= \int \exp(\lambda xy)dP_{(X,Y)}(x,y) \tag 1 \\ &= \int \int \exp(\lambda xy)dP_X(x) dP_Y(y) \tag 2\\ &= \int \left(\int \exp(\lambda xy)dP_Y(y)\right) dP_X(x) \tag 3\\ &= \int E(\exp(\lambda xY)) dP_X(x)\\ &= \int \exp( \lambda^2 x^2)/2\; dP_X(x)\\ &= E(\exp( \lambda^2 X^2)/2) \end{align}$$
$(1)$: law of the unconscious statistician
$(2)$: independence of $X$ and $Y$
$(3)$: Tonelli's theorem