I am looking at the various ways to derive explicit position-space expressions of the Green's function with certain boundary conditions (Feynman propagator) of the Klein-Gordon equation. In this particular example, I cannot figure out why, when integrating by parts, the boundary term, i.e., the $uv$ in $\int u dv = u v - \int v du$, disappears. The result of the whole process is, as usual, a modified Bessel function. I know there are multiple different methods to perform this integration; I understand these and am not interested., i.e., through contour integration, or Schwinger parameterization, I specifically want to know why, on the second line on the below, that we discard the evaluated term from the IBP?
Whenever I try I get something like
\begin{equation} uv = \left[ \left( -\frac{k}{\sqrt{k^2 + m^2}} \right) \cos kr \right]^{\infty}_{0} = [-1\cdot \cos\infty] + 0. \end{equation}