Product of two random variables that has exponential distribution

Ok, here goes.

Assuming $X_1,X_2$ are independent and $Y=X_1X_2$ we have

\begin{multline} P(Y\leq y)= P(X_1X_2\leq y) = \int_0^\infty \lambda P(X_1\leq y/x) e^{-\lambda x} dx = \lambda \int_0^\infty (1-e^{-\lambda y/x}) e^{-\lambda x} dx =\\ 1- \lambda \int_0^\infty e^{-\{\lambda (x+y/x)\}} dx. \end{multline}

The density is the derivative of that with respect to $y$, i.e. \begin{equation} \lambda^2 \int_0^\infty \frac{1}{x} e^{-\{\lambda (x+y/x)\}} dx, \end{equation} for all $y>0$.

The density of $X_1 X_2$, obtained from that integral others have posted, is $f(z) = 2 \lambda^2 K_0(2 \lambda \sqrt{z})$ for $z > 0$, $0$ otherwise, where $K_0$ is a modified Bessel function of the second kind.