Adapted from L. H. Ryder Quantum Field Theory

Given a diagonal matrix \(A=diag(a_1,a_2,\dots,a_n)\), the quadratic (exponent) integral $$ \int e^{\frac{1}{2}\textbf{x}A\textbf{x}} d^n x= (2\pi)^{n/2} (\det A)^{-1/2} $$