indeed, integration by parts it is:

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

where I have used that $\mathbf{C}$ is a symmetric matrix