Case 2:

$$\int \exp\left( - (Y^{\mathrm{\textsf{T}}}A Y + B Y + C) \right) \; dY = \int \exp\left( - (Z^{\mathrm{\textsf{T}}}A Z + C - \frac{1}{4} B A^{-1} B^{\mathrm{\textsf{T}}}) \right) \; dZ$$

$$= \left(\pi\right)^{N/2} \vert\det(A)\vert^{-1/2} \exp\left( \frac{1}{4} B A^{-1} B^{\mathrm{\textsf{T}}}- C \right)$$ (16)

where $Z = Y + \frac{1}{2} A^{-1} B^{\mathrm{\textsf{T}}}$ and $A^{-1}$ exists.