How would I go about obtaining the probability density function of a random variable that results in a product of a discrete variable and a continuous variable? I know that if $X$ and $Y$ are both continuous, then the probability density function of $Z=XY$ is given by: $$f_Z(z)=\int_{-\infty}^{\infty}f_X(x)f_Y\left(\frac{z}{x}\right)\frac{1}{|x|}dx$$ but what is the method of obtaining $f_Z(z)$ if $X$ is a discrete variable and $Y$ is continuous?
More specific to my case, I have $X$ following a Rademacher distribution, that is $$\begin{align*} P(X=x) = \begin{cases} \frac{1}{2} & x = -1\\ \frac{1}{2} & x = 1\\ 0 & otherwise \end{cases} \end{align*}$$ and $Y$ following a Rayleigh distribution. What would be the product of these two random variables?