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I found a formula around a Wallis-like integral:

Let $\displaystyle I_n = \int^{\frac{\pi}{2}}_{0} x \sin^n x dx $ for $n \in \mathbb{N_{\geq0}}$,

then:

$$I_{n+2} = \frac{n+1}{n+2} I_n + \frac{1}{(n+2)^2}$$

and $\displaystyle I_0 = \frac{\pi^2}{8}, \quad I_1 = 1, \quad nI_n^2 \xrightarrow{n \to \infty} \frac{\pi^3}{8}$, therefore we can obtain:

$$\sum^{\infty}_{n=0} \frac{I_n}{n+1} = \frac{\pi^3}{8}$$

My question here is that: Are there any interesting series involving "higher-degree" wallis-like integrals?

(For example, $J_n = \displaystyle \int^{\frac{\pi}{2}}_{0} x^2 \sin^n x dx$ is one of "higher-degree" wallis-like integrals).

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