I am trying to solve, $$I=\int_{0}^{1} x^2 J_0(k_nx) \ dx.$$
My attempt:
I choose make the substitution $u=k_nx$, which has led to, $$I=\frac{1}{k_n^3}\int_{0}^{k_n} u^2 J_0(u) \ du.$$ Looking at an integral table, I see that $$\int x^2J_0(x) \ dx=x^2J_1(x)-\phi(x) \ \ \text{where} \ \ \phi(x)=\frac{\pi x}{2}\left(J_1(x)\cdot H_0(x)-J_0(x)\cdot H_1(x)\right).$$ Unfortunately, I think this is beyond the scope of my course, as I have not heard of the Sturve function $(H_v(x))$.
Am I missing something simple here?
As the problem is set, I do not see how you could avoid Struve functions for the integral.
For the antiderivative, what you could use is $$I=\int u^2 J_0(u) \, du=\frac{u^3}{3} \, _1F_2\left(\frac{3}{2};1,\frac{5}{2};-\frac{u^2}{4}\right)$$ which of no much interest since it simplifies to $$I=\frac{1}{2} u ((2 u-\pi \pmb{H}_0(u)) J_1(u)+\pi \pmb{H}_1(u) J_0(u))$$ you already found in the integral table.