I have a question about a integral that have singularity at the endpoint. The singularity will bring oscillation to the solution. I am wondering if there is a way to approximate this integrand with a recurrence relation so that we can smoothe out the oscillation? Here is the integral.

integral exp(-k*L) /((Ro*k)^2 + Ro^2*(U*k-1)^2-1) *(exp(-Ro*k*z/(1-Ro^2*(U*k-1)^2)^0.5 ) *exp(i(kx-t)) dk from (Ro-1)/Ro/U to (Ro+1)/Ro/U. Here Ro(=0.707), U(=0.625), t(=0.25) are const; we can set x, z as arrays from -10 to 10 and from 0 to 10. As we notice that the singularity exists at endpoint (Ro+1)/Ro/U. My question is if it is possible to modify the integral to a suitable form which can smoothe out the singularity? For example, if we can simplify it to the form as integral exp(t)/t dt from const to infinity, it will be solvable.

Thanks.