I am trying to show that
$$ \int J_0(x) \cos(x) dx = x J_0 \cos(x) + x J_1 \sin(x) +C$$
where $J_n$ is the Bessel function of the first kind, using integration by parts. However, both obvious factors lead nowhere.
$$ \int J_0(x) \cos(x) dx = J_0 \sin(x) + \int \sin(x) J_1(x) dx + C$$
using $(J_0(x))'=-J_1(x)$. Also, the integral $\int J_0(x) dx$ is not known. Any hints? Thanks!
Best Answer
As Daniel Fischer pointed on the comment above, the integral can be computed as follows:
$\int J_o(x) \cos(x) dx=x J_0(x) \cos(x) + \int x J_1(x) \cos(x) dx + \int x J_0 \sin(x) dx +C= J_0(x) \cos(x) + \int x J_1(x) \cos(x) dx+x J_1(x) \sin(x) - \int x J_1(x) \cos(x) dx$
which produces the desired result.