# Finite Integral involving Bessel Function, $J_1$

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.

• Ask Wolfram Alpha. – Yves Daoust Apr 24 '16 at 20:13
• Thanks, @YvesDaoust, I don't know why I didn't check that myself in the first place. I am not familiar with Struve Functions, so if anyone has a different way of getting to the solution, it would be appreciated. – Julien V Apr 25 '16 at 13:26

$$\int_0^a dx \ x \ J_1(x) = \frac{\pi a}{2} \left( H_0(a)J_1(a) - H_1(a)J_0(a) \right) \ .$$
$H_n(x)$ is the Struve Function.
• For even $n$, you can get the integral $\int x J_n(x)\;dx$ not using Struve. – GEdgar Apr 25 '16 at 13:38