TITLE: How to solve $\int_0^\infty J_0(x)\ \text{sinc}(\pi\,x)\ e^{-x}\,\mathrm dx$?
QUESTION [26 upvotes]: I need some help with solving this integral involving Bessel function:
$\hspace{2in}\displaystyle\int_0^\infty$$J_0(x)\ $$\text{sinc}(\pi\,x)\ $$e^{-x}\,\mathrm dx.$

REPLY [16 votes]: Here's an evaluation of the integral used by O.L..
$$\begin{align} \int_{0}^{\infty} \frac{1-e^{-px}}{x} J_{0}(x) \ dx &= \int_{0}^{\infty} \int_{0}^{\infty} (1-e^{-px})J_{0}(x) e^{-xt} \ dt \ dx  \\ &= \int_{0}^{\infty} \left( \int_{0}^{\infty} J_{0}(x) e^{-tx} \ dx \ - \int_{0}^{\infty} J_{0}(x) e^{-(p+t)x} \ dx \right) \ dt  \\ &= \int_{0}^{\infty} \Bigg( \frac{1}{\sqrt{1+t^{2}}} - \frac{1}{\sqrt{1+(p+t)^{2}}} \Bigg)\ dt \tag{1} \\ &= \text{arcsinh}(t) -\text{arcsinh}(p+t) \Bigg|^{\infty}_{0} \\ &= \text{arcsinh}(p) \\ &= \ln \left(p+\sqrt{p^{2}+1} \right) \end{align}$$
$(1)$ http://mathworld.wolfram.com/LaplaceTransform.html