Lommel's Integrals

$$(\beta^2-\alpha^2) \int xJ_n(\alpha x)J_n(\beta x)\,dx = x[\... ...a J_n'(\alpha x)J_n(\beta x)-\beta J_n'(\beta x)J_n(\alpha x)]$$

$$\int x{J_n}^2(\alpha x)\,dx = {\textstyle{1\over 2}}x^2[{J_n}^2(\alpha x)+J_{n-1}(\alpha x) J_{n+1}(\alpha x)],$$

where $J_n(x)$ is a Bessel Function of the First Kind.

References

Bowman, F. Introduction to Bessel Functions. New York: Dover, p. 101, 1958.