Let $X_1$ and $X_2$ two independent random variables. The PDF of $X_i$ is $f_{X_i}(x_i)$.
Ho we can find the following expectation $$E\exp(X_1+X_2)\mathbf{I}_{\{X_1<X_2\}}.$$
Is it
\begin{align} E\exp(X_1+X_2)\mathbf{I}_{\{X_1<X_2\}}&=Ee^{X_1}e^{X_2}\mathbf{I}_{\{X_1<X_2\}}\\ &=\int_{x_2=-\infty}^{\infty}e^{x_2}f_{X_2}(x_2)\left(\int_{x_1=-\infty}^{x_2}e^{x_1}f_{X_1}(x_1)dx_1\right)dx_2. \end{align} Thanks.