I am trying to reproduce a paper and there is a term in it, $x_{nm}$, which is given as : $$ x_{nm} = \int_{0}^{\frac{1}{\sqrt{\pi}}} r dr \int_{0}^{2\pi} d\theta \ \psi_{nk_{1}l}^{*} \ r\ cos(\theta)\ \psi_{mk_{2}l} $$ where, $$ \psi_{nkl} = \frac{\pi}{J_{k+1}(\rho_{kl})} J_k (\rho_{kl}) e^{ik\theta} $$ and $\rho_{kl}$ represents $l$-th root of the Bessel Function of first kind, s.t. $J_k(\rho_{kl})=0$.
Since, $J_k(\rho_{kl})$ does not depend on $\theta$, I evaluate $x_{nm}$ :
$$
x_{nm} \sim \int_{0}^{2\pi} d\theta \ e^{i(k_2-k_1)\theta} \ cos(\theta) = \frac{-i (k_2-k_1) (e^{2i\pi(k_2-k_1)}-1)}{(k_2-k_1)^{2} - 1}
$$
The integral is checked using Mathematica and WolframAlpha. Now, it is a requirement that $k_i \ \in \ \mathbb{Z}$, the set of integers. So, it appears that $x_{nm}$ will always be zero but infact according to the paper mentioned above it is not.
I must be going wrong somewhere. Can someone please help me out?