I have a confusion over an integral involving a multivariate and a univariate Gaussian. We know that in the case of two multivariate Gaussians the following is true: $$ \int \mathcal{N}(\mathbf{y}|\mathbf{A}\mathbf{x} + \mathbf{b}, \mathbf{S}) \mathcal{N}(\mathbf{b}|\mathbf{m}, \mathbf{C}) \mathbf{db} = \mathcal{N}(\mathbf{y}|\mathbf{A}\mathbf{x} + \mathbf{m}, \mathbf{S} + \mathbf{C}) $$
Could somebody please tell me what the integral evaluates to in the following case where $b$ is now a scalar: $$ \int \mathcal{N}(\mathbf{y}|\mathbf{A}\mathbf{x} + b\mathbf{1}, \mathbf{S}) \mathcal{N}(b|m, c^2) db $$
where
- $\mathbf{1}$ is a column of ones of the same dimension as $\mathbf{y}$
- $\mathcal{N}(b|m, c^2)$ is the univariate Gaussian distribution with mean $m$ and standard devation of $c$.