Integration in path integral formalism?

[physengineer]

Hi,

Does anyone know how this integral is calculated

$$\int[dx] x_i x_j \exp \{ - (\frac{1}{2} \sum_{rs} A_{rs}x_r x_s+\sum_r L_r x_r ) \}$$

Thanks

 

[fzero]

First note that

$$\int[dx] x_i x_j \exp \{ - (\frac{1}{2} \sum_{rs} A_{rs}x_r x_s+\sum_r L_r x_r ) \} = \frac{\partial}{\partial L_i} \frac{\partial}{\partial L_j} \int[dx]\exp \{ - (\frac{1}{2} \sum_{rs} A_{rs}x_r x_s+\sum_r L_r x_r ) \} .$$

Now the integral on the RHS above can be performed by completing the square in the exponent.