I have to evaluate the following two integrals that I would like to solve,
Sin[θ] 2 π BesselJ[0, k Sin[θ] (μ[j] - μ[i])] Exp[2 k^2 σ^2 Cos[θ]^2],
and
Sin[\[Theta]]^2 2 I \[Pi] BesselJ[1, k Sin[\[Theta]] (\[Mu][j] - \[Mu][i])] Exp[2 k^2 \[Sigma]^2 Cos[\[Theta]]^2]
w.r.t. $\theta$ between the interval [0, $\pi$]. The first equation consists of a zeroth-order Bessel function and the second a first-order Bessel function. These equations arise by evaluating the inner product of a wavefunction which resembles a collection of 2D Gaussian state along the y-axis with centres at $\mu[i]$.
I have tried to evaluate the integral using the Mathematica sybolic language but with no results. Can a suitable transformation be suggested to achieve an analytic solution to both equations or a solution by any other means?
Thanks very much for your help!
Note, I posted this question in Mathematica Stack exchange and was advised to ask on the Mathematics Stack exchange.