The following exercise takes the form;
$\int_{0}^{\infty}f\left(\frac{P_{n+1}\left(x\right)}{P_{n}\left(x\right)}\right)\cdot\frac{1}{P_{n}\left(x\right)^{2}}dx=\left(n+1\right)\int_{0}^{\infty}f\left(u\right)du$
How is this so?
For the nth Legendre Polynomial, $P_n(x)$. It looks to be a u-substitution of the rational function, with the denominator of du already in place, but the numerator is not quite there to complete the sub. Suspiciously the part that is missing is almost a Wronskian between the two functions, but I am not sure how to proceed or if this is useful.
Any ideas for completing the transformation would be much appreciated; thanks!