Integration of first kind zero order bessel function J(x)

Question

What is the integration of first kind zero order Bessel function $J(kx)$, where $k$ is constant and the limits of $x$ are from $0$ to $a$.

i.e., $$\int_0^{a} J_0(kx) dx = ?$$

Answer 1

If you look here, you wil find that

$$\int J_0(t)\,dt =\frac{\pi t}{2} \pmb{H}_0(t) J_1(t)+\frac{t}{2} (2 -\pi \pmb{H}_1(t)) J_0(t)$$ where appears the Struve function.

This makes $$\int_0^a J_0(kx)\,dx =\frac{a}{2} (\pi \pmb{H}_0(a k) J_1(a k)+(2-\pi \pmb{H}_1(a k)) J_0(a k))$$