I'm looking at a paper using Biot-Savart and Ampere's law to determine the induced magnetic field within a conducting cylinder. By inserting
$$J_z = \frac{I}{2\pi} \int_0^\infty \lambda J_0(\lambda r) exp(-\lambda^2\sigma_j^2/4d) \frac{\sinh(\lambda(c-z))}{\sinh(\lambda c)}d\lambda$$
into
$$B_\theta = \frac{\mu_m}{r} \int_0^r r J_z dr $$
they obtain
$$B_\theta = \frac{\mu_mI}{2\pi} \int_0^\infty J_1(\lambda r) exp(-\lambda^2\sigma_j^2/4d) \frac{\sinh(\lambda(c-z))}{\sinh(\lambda c)}d\lambda$$
I'm having troubles to understand how they obtained the final expression for $B_\theta$. Can anyone help and explain it step by step?
Here, $B_\theta$ and $J_z$ are the magnetic flux and current density components in an axisymmetric domain (r, z, $\theta$). $\sigma_j$, $\mu_m$, c and d are constants, and $J_0$ and $J_1$ are Bessel functions.