Do you guys have any ideas on solving the following definite integral: $$\int_{0}^{R} x^{\lambda} (1-x)^{\mu-1} K_{1}(a\sqrt{x}) \, dx$$ where the parameters $\lambda>0$, $0 < \mu <1$, $a>0$, and $K_{1}(\cdot)$ is the modified Bessel function of second kind with order 1.
I tried to conduct the integration by parts but eventually the (1-x) term will just not vanish. I also found the following definite integral from the book `Table of Integrals, Series and Products´. However, it is not straightforward for me to derive the antiderivative from the definite integral
