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Notation: $\mathrm{y}_l(x)$ is the spherical Bessel function of second kind.

I need to calculate the following indefinite integral: $$\int x^2\left(\mathrm{y}_l\left(x\right)\right)^2dx$$

I tried it using the differential equation itself and got no result. I have looked up the standard book of tables and DLMF to no avail. I'm looking for any suggestions to obtain a closed-form expression. Thanks in advance!

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  • $\begingroup$ CAS says: $\int x^2 y_l(x){}^2 \, dx=\frac{1}{4} \pi x^2 Y_{\frac{1}{2}+l}(x){}^2-\frac{1}{4} \pi x^2 Y_{-\frac{1}{2}+l}(x) Y_{\frac{3}{2}+l}+C$ where:$Y_l(x)$ is Bessel function of the second kind. $\endgroup$ Jun 19 '18 at 21:50
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Exactly the same approach works as I gave in my answer to your previous question, except that this time the Wronskian terms vanish, so you obtain $$ \int x^2 (y_l(x))^2 \, dx = \frac{x^3}{4}(y_l''(x)y_l(x)-y_l'(x)y_l(x))+C, $$ which again can be simplified if desired.

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