Find the pdf of the product of the two i.i.d standard normal.
I have seen several similar questions but none of them was the derivation of the product's pdf.
$X,Y \sim \mathrm{Norm}(0,1)$
$XY=U$, let $Y=V$ and $X = U/V$
$$f_U(u) = \int f(x=u/v, y = v)dv = \int_{-\infty}^{\infty} f (x=u/v)f(y=v)dv/|v|$$ where the $1/|v|$ part comes from the Jacobian matrix after doing partial derivatives by $U$ and $V$ of $X$ and $Y$, combining them in the matrix and computing its determinant.
$$f_U(u) = 1/\pi \int_{0}^{\infty}e^{(u/v)^2/2}e^{-v^2/2}dv/v=1/\pi \int_{0}^{\infty}e^{u^2/2v^2-v^2/2}dv/v$$
How to solve the integral? I thought about substituting $t=u/v$ and using $dt=-u/v^2dv$ but that leads me nowhere.
The information provided by @Eric Towers in the comments was sufficient.