Integration of mixture distribution

Question

Suppose $y | \mu \sim N(\mu, \sigma^2)$, $\mu \sim w\delta_0 + (1-w)N(m_0, s_0^2)$, where $\delta_0$ is Dirac delta function(In bayesian statistics, this mixture is called spike and slab). Now I want to find distribution of $y$.
My solution is
$f(y) = \int p(y|\mu) p(\mu) d\mu = w * N(y|0, \sigma^2) + (1-w)*\int N(y|\mu, \sigma^2) * N(\mu | m_0, s_0^2) d\mu$.
Am I right?