Definite integrals involving product of spherical bessel functions of the first kind

Question

I'd like to solve the following integrals $$\int_0^R |J_n(x)|^2 xdx$$ where $J_n(x)$ is the $n$-th order Bessel function of the first kind, and $R$ is a positive real constant.

$$\int_0^R |j_n(x)|^2x^2dx$$ where $j_n(x)$ is the $n$-th order spherical Bessel function of the first kind, and $R$ is a positive real constant.

Any ideas on how to tackle these problems?

Thanks a lot!

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I find the first integral at Integral of product of Bessel functions of the first kind, so any ideas about the second integral?

Answer 1

Using the identity $$j_n(x)=\sqrt{\frac{\pi }{2x}} J_{n+\frac{1}{2}}(x)$$ $$\int x^2 \,j_n(x){}^2 \,dx=\frac{\pi}{2} \int x J_{n+\frac{1}{2}}(x){}^2 \,dx=\frac{\pi}{4} x^2 \left(J_{n+\frac{1}{2}}(x){}^2-J_{n-\frac{1}{2}}(x) J_{n+\frac{3}{2}}(x)\right)$$