# Wallis-like integrals

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).