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Are you interested in Gaussian integrals themselves or their role in some particular part of QFT? If it's the former, there's really not much to it--a "Gaussian integral" is just an integral of the exponential of a quadratic, usually understood to be over the real line. Every quadratic represents a parabola, which can be written in vertex form, giving the general (1-D) Gaussian integral as I = Int_R[ exp(-λ(x-α)²+β] dx ], λ>0 for convergence. By translational symmetry, we have I² = Int_R²[ exp(-λ(x²+y²)+2β) dx dy ] = exp(2β) Int_R²[ exp(-λr²) r dr dθ ] = 2π exp(2β) Int_0^infty[ exp(-λr²) r dr ], so that I = e^β [π/λ]^{1/2}. Polar coordinates makes things very simple; recall differentiating {x = r cos θ, y = r sin θ} gives dx dy = r dr dθ.

Now, generalize to R^n: I = Int_{R^n}[ exp{-x'x/2} dx ] = [2π]^{n/2}, where ' denotes matrix transpose and x is interpreted as a column vector. The proof is rather obvious from the above, being just an n-ary integral of independent Gaussians in Cartesian coordinates. Now, suppose that A is an invertible n×n matrix; then the transposrmation x→Ax gives I_V = Int_{R^n}[ exp(-/2} dx ] = [2π det(V)]^{n/2}, V = AA', U = inv(V), = x'y (hint: the Jacobian is trivial).

For the purposes of QFT, most books will assume that the matrix is positive-definite as well; however, starting with an arbitrary real A, AA' is always positive-definite, and this approach illuminates the reason behind the general formula a little better (it may be helpful to look into probability, cf. multinormal/multivariate gaussian distribution especially). Furthermore, for the purposes of QFT, the given matrix will have to be complex, but that's not really an issue, as it can always be split into real and imaginary parts. The important part is that Re V is positive-definite. It's essentially the same type of thing, just recast in a different context (e.g., dz* dz = dx dy, norm involving conjugate, etc.).

I hope that was ordinary Gaussian integration that was covered in class so far, at least. If it was generalized Gaussian integration (with Grassman variables and Berezians instead of determinants), you have a bit more catching up to do, even if it is analogous.