First-passage time density for a level x above zero

A standard Wiener process starts at $0$. Let $T_x$ be the first time it reaches the level $x>0$. Find the probability density of $T_x$ and its expected value.

$f(t)=\dfrac{x}{\sqrt{2\pi t^3}}e^{-x^2/(2t)}$ for $t>0$, and $\mathbb{E}[T_x]=\infty$. The reflection principle gives $\mathbb{P}[T_x\le t]=2\,\mathbb{P}[W_t\ge x]=2\big(1-\Phi(x/\sqrt t)\big)$; differentiating in $t$ yields the density. The level is hit with probability $1$, but the density decays only like $t^{-3/2}$, so the mean is infinite.