Skewness of a shifted Laplace distribution

A continuous random variable has density $\tfrac{\lambda}{2}e^{-\lambda|x-\mu|}$ on the real line, with constants $\lambda>0$ and $\mu>0$. Assuming its first three moments are finite, find its skewness $\mathbb{E}\left[\left((X-\mu_X)/\sigma_X\right)^3\right]$.

$0$. The density is symmetric about $\mu$, so $\mu_X=\mu$ and the third central moment is the integral of an odd function, $u^3e^{-\lambda|u|}$, over the whole line, which vanishes.