Solve using Bessel function properties

Question

I am trying to calculate the value of $$ \int{x^3J_3(x)dx}$$ I know I have to manipulate the recurrence relation properties of bessel equation like : $$ x^nJ_n(x) = \int{x^nJ_{n-1}(x)}$$ and $$x^{-n}J_n(x) = \int{-x^{-n}J_{n+1}(x)}$$ . But I'm not able to foresee how these will lead to what I want to calculate. Hint about how to start will do and I have to solve it using bessel properties only.

Answer 1

The question should have options or should have stated it want answer in terms of bessel function. $$\int{x^3J_3(x)dx} = \int{x^5x^{-2}J_3dx}$$ $$ =x^5(-x^{-2}J_2) - \int{5x^4.(-x^{-2}J_2)dx}$$ $$ =-x^3J_2 + 5\int{x^2J_2dx}$$ $$ =-x^3J_2 + 5\int{x^3.(x^{-1}J_2dx}$$ $$ =-x^3J_2 +5x^3.(-x^{-1}J_1) -5\int{3x^2.(-x^{-1}J_1)dx}$$ $$ =-x^3J_2 -5x^2J_1 +15\int{xJ_1dx}$$ $$ = -x^3J_2 -5x^2J_1 +15x.(-J_0) -15\int{-J_0dx}$$