How to Solve the Integral of x^3*J3(x)?

[jaron_denson]

Hello Everyone trying to come up with a stratagey to solving this integral

Int(x^3*J3(x),x) no limits

Ive tried some integration by parts and tried breaking it down into J1 and J0's however i still get to a point where I have to integrate either : Int(x*J1(x),x) or Int(J6(x),x)

 

[jaron_denson]

 

[Astronuc]

Use the following recursion relationships.

Start with the first one, and let n+1 = 3

$$\frac{2n}{x}\,J_n(x)\,=\,J_{n-1}(x)\,+\,J_{n+1}(x)$$

$$2\frac{dJ_n(x)}{dx}\,=\,J_{n-1}(x)\,-\,J_{n+1}(x)$$

$$\frac{dJ_0(x)}{dx}\,=\,-J_1(x)$$

Of course, one could use the more general derivative identity

$$\frac{d}{dx}[x^m J_m(x)]\,=\,x^m J_{m-1}(x)$$

but one should probably prove that.

http://mathworld.wolfram.com/BesselFunctionoftheFirstKind.html