indeed, integration by parts it is:

$$\int d\vec{x}\; \vec{x} \,f(\vec{x})\exp\left(-\tfrac{1}{2}\vec{x}\cdot C^{-1}\cdot\vec{x}\right)=-\int d\vec{x}\;  \,f(\vec{x})\,C\cdot\frac{\partial}{\partial \vec{x}}\exp\left(-\tfrac{1}{2}\vec{x}\cdot C^{-1}\cdot\vec{x}\right)$$
$$=\int d\vec{x}\;  \,\exp\left(-\tfrac{1}{2}\vec{x}\cdot C^{-1}\cdot\vec{x}\right)C\cdot\frac{\partial}{\partial \vec{x}}\, f(\vec{x})$$

where I have used that $C$ is a symmetric matrix