(**Edit:** Thank you Vladimir for providing the references for the closed form value of the integrals. My revised question is then to how to derive this closed form.)

For all $n\in\mathbb{N}^+$, define $\mathcal{I}_n$ by the definite integral, $$\mathcal{I}_n:=\int_{0}^{\infty}\frac{\sin^n{(x)}}{x^n}\mathrm{d}x.$$ Prove that $\mathcal{I}_n$ has the following closed form: $$\mathcal{I}_n\stackrel{?}=\pi\,2^{-n}\left(n\sum_{k=0}^{\lfloor\frac{n-1}{2}\rfloor}\frac{(-2)^k(n-2k)^{n-1}}{k!(n-k)!}\right),~~\forall n\in\mathbb{N}^+.$$

Integrals of small positive integer powers of the $\operatorname{sinc}$ function come up on a regular basis here, but it occurred to me that while I probably know the derivations for the $1\le n\le 4$ cases like the back of my hand, I can't recall ever working the integrals wfor any value of $n$ higher than that. The values of the first four integrals are,

$$\mathcal{I}_1=\frac{\pi}{2},\\ \mathcal{I}_2=\frac{\pi}{2},\\ \mathcal{I}_3=\frac{3\pi}{8},\\ \mathcal{I}_4=\frac{\pi}{3}.$$

So I set out to first calculate $\mathcal{I}_5$ to see if any obvious pattern jumped out (and see if the trend of being equal to rational multiples of $\pi$ continued). I wound up getting frustrated and asking WolframAlpha instead. It turns that while the first four cases hinted very much at the possibility of a simple pattern relating the values of $\mathcal{I}_n$ for different positive integers $n$ (or possibly two separate patterns for even and odd $n$), the next few values most definitely did not:

$$\mathcal{I}_5=\frac{115\pi}{384},\\ \mathcal{I}_6=\frac{11\pi}{40},\\ \mathcal{I}_7=\frac{5887\pi}{23040}\\ \mathcal{I}_8=\frac{151\pi}{630}.$$

So my questions are, 1) is there a systematic way to compute these integrals for all $n$?; and 2) is there an elegant way to represent these values in closed form for general $n$?