I have the following equation: $I(k,x)=\int_0^xJ_k(\tau^k)d\tau=\alpha$ where $\alpha$ is a given constant $\alpha\in \mathbb{R}$ and $k$ integer with $k\gt 0$. $J_k(x)$ is the Bessel function of first kind. My question is: is it possible to solve the equation for every $k$ or there is only a couple $(k_0,x_0)$, solving the $I(k,x)=\alpha$?
Thanks in advance for suggestions