Consider the following integral $$ pv\int_0^{\infty}e^{N(-2Ax+A\log x)}\frac{e^{-B\log x}}{1-2x}dx $$ where $A,B>0$ and we take the Cauchy principal value at $x=1/2$. I am interested in obtaining the asymptotics when $N$ is very big. The first thing I thought of was some variant of Laplace's method but I am unsure if I can proceed here, because of the singularity at $x=1/2$. So, my question is, is some version of the Laplace's method applicable here to obtain the big $N$ asymptotics? and if so, how should I proceed?