Finite Integral involving Bessel Function, $J_1$

Question

Is there a closed form solution for the following integral

$$ \int_0^a dx \ x \ J_1(x) \ , $$ where $J_1(x)$ is a Bessel Function and $a>0$ is a finite real number?

Thanks.

Answer 1

The solution, according the Mathematica, is

$$ \int_0^a dx \ x \ J_1(x) = \frac{\pi a}{2} \left( H_0(a)J_1(a) - H_1(a)J_0(a) \right) \ . $$

$H_n(x)$ is the Struve Function.