How to use saddle point approximation with path integrals?

Question

i would like to evaluate $$\int\mathcal{D}x\ e^{-\int\limits_{-\infty}^{\infty} dt\ (\dot x+\alpha x)^2}\tag{1}$$ and it is my understanding that the way to do so is using the saddle point approximation, however I'm new to using it and am unsure how to do so for the case of an action

i was able to solve the Euler Lagrange equations for $$\mathcal{L}=(\dot x+ \alpha x)^2\tag{2}$$ however i am not sure of the next step

usually i would Taylor expand the function inside the exponent and evaluate the resulting gaussian integral but i am not sure if this requires me to take the second derivative of my action with respect to $x$ (or $\dot x$ as well?)

how do i proceed?

Answer 1

The crossterms in OP's Lagrangian (2) are total derivative terms, so OP's partition function (1) with Dirichlet boundary conditions is essentially an ordinary harmonic oscillator (in Euclidean signature), which is e.g. done exactly in this Phys.SE post. The exact result coincides with the saddle point approximation.